Essays in Spatial Econometrics - AU Pure · econometrics have provided a powerful tool for examining spatial dynamics across different economic units. In contrast to standard econometrics,

Essays in Spatial Econometrics

2016-8

Girum Dagnachew Abate

PhD Thesis

DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS

AARHUS UNIVERSITY � DENMARK

ESSAYS IN SPATIAL ECONOMETRICS

BY GIRUM DAGNACHEW ABATE

A PhD thesis submitted to

School of Business and Social Sciences, Aarhus University,

in partial fulfillment of the requirements of

the PhD degree in

Economics and Business Economics

January 2016

PREFACE

This thesis was written in the period from February 2013 to January 2016 during my

PhD studies at the Department of Economics and Business Economics at Aarhus

University. During my graduate studies, I was affiliated with the Center for Research in

Econometric Analysis of Time Series (CREATES), funded by the Danish National

Research Foundation. I would like to thank the Department of Economics and

Business Economics, Aarhus University and CREATES for providing me with inspiring

and excellent research environments and the financial support throughout my studies.

This has made it possible for me to attend various courses, workshops and

conferences both nationally and internationally.

I would like to take the opportunity to thank a number of people. First and

foremost, I would like to thank my main supervisor Professor Bent Jesper Christensen

for his continued encouragement and practical guides during my graduate studies and

encouraging me to enroll as a PhD student after my MSc study at Aarhus University. I

am grateful to my co-supervisor and co-author Professor Niels Haldrup for his support

and useful advice on research projects as well as my career path. I sincerely appreciate

your interesting ideas and commitment during our joint research work which is

included as a third chapter in this dissertation. Working with you has been an inspiring

and great learning experience.

From January 2015 to May 2015, I was fortunate to visit Professor Luc Anselin at the

GeoDa Center for Geospatial Analysis and Computation, School of Geographical

Sciences and Urban Planning, Arizona State University. I would like to thank Luc for

inviting me, for the inspiring and interesting discussions we had during our joint work

and his invitations to attend some seminars at the Economics Department of Arizona

State University and I am looking forward for more joint works in the near future. I

would also like to thank Professor Sergio Rey for the interesting and important

discussions we had during my stay at Arizona State University. I am grateful to Dr. Julia

Koschinsky and Dr. Robert Pahle for making my stay at Arizona pleasant and easy. I am

thankful for GeoDa center staff, faculty members and graduate students for the

welcoming atmosphere. I enjoyed the welcoming environment at Arizona State

i

ii

University.

I am thankful for all my PhD fellow students and other colleagues at the

Department of Economics and Business Economics for the friendly and enjoyable

atmosphere. I am grateful to Solveig Nygaard Sørensen, CREATES’ administrator for

proofreading of the thesis and for all the practical help and conversations in the last

three years. Administrative support to Head of the PhD unit (Aarhus University),

Susanne Christensen, deserves a special thank for always being there for practical

helps in relation to my PhD activities.

Finally, I would like to thank all my family for all their love and encouragement.


Aarhus, January 2016

UPDATED PREFACE

The pre-defence meeting was held on March 15, 2016 in Aarhus. I would like to

thank the assessment committee consisting of Professor David Edgerton, Lund

University, Professor Jørgen Lauridsen, University of Southern Denmark, and

Associate Professor Morten Berg Jensen (chair), Aarhus University for their careful

reading of the thesis, and for the constructive and insightful comments and

suggestions. Some of the suggestions have been incorporated in the present version of

the dissertation, while others remain for future work.


Arizona, March 2016

i

CONTENTS

Contents iii

Summary vi

Danish Summary xi

1 On the link between volatility and growth: A spatial econometrics approach 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 The empirics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Non-spatial Ramey-Ramey model . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 The spatial Ramey-Ramey model . . . . . . . . . . . . . . . . . . . . . . . 91.3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.5 Direct and indirect effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Alternative regression frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 House price fluctuations and macroeconomic dynamics 312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Brief literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 Empirical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.2 Direct and indirect impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5.1 Dynamic panel analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.5.2 Spatial modeling of house prices and the macroeconomic dynamics 482.5.3 Alternative regression frameworks . . . . . . . . . . . . . . . . . . . . . . 49

2.5.3.1 Direct and indirect impacts . . . . . . . . . . . . . . . . . . . . . 502.5.3.2 MSA fixed effects specification . . . . . . . . . . . . . . . . . . . 51

2.5.4 Time varying space-time model results . . . . . . . . . . . . . . . . . . . 522.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Space-time modeling of electricity spot prices 643.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 The Nordic Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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CONTENTS iv

3.3 Spatial modeling of spot prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 Data description and spatial weight matrices . . . . . . . . . . . . . . . . . . . . . . 74

3.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4.2 Spatial weight matrix for spot prices . . . . . . . . . . . . . . . . . . . . . 77

3.5 Estimation results and forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5.1 Quasi-maximum likelihood estimation of the SDM . . . . . . . . . . . . 813.5.2 Empirical results and test for spatial interaction effects . . . . . . . . . 823.5.3 Direct and indirect effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.5.4 Forecasting performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.5.5 A time-varying coefficients SDM . . . . . . . . . . . . . . . . . . . . . . . 90

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

SUMMARY

This thesis comprises three self-contained chapters on the spatial econometric

analysis of cross-sectional and panel data problems that exhibit a clear spatial

dimension. With increasing market integration and globalization, recent theoretical

and empirical works focus on the interdependence of economic units (countries,

market areas and metropolitan areas etc.) indicating that the economy of one unit is

not independent of the economies of others. Recent developments in spatial

econometrics have provided a powerful tool for examining spatial dynamics across

different economic units. In contrast to standard econometrics, in spatial

econometrics each space (location) is explicitly modeled in model estimation and

testing.

In terms of model specification and estimation, there are two broad strands of the

spatial econometrics literature. The first strand of the literature focuses on the

cross-sectional spatial lag specification where spatial dependence is accounted

through lags in the spatial dimension (Anselin 1988). The second strand of the

literature focuses on spatio-temporal specification where both the spatial dimension

and the temporal dimension are explicitly incorporated in model estimation and

testing (Elhorst 2012). The various chapters in this thesis contribute to these two

different strands of the literature with a broader theme of spatial econometric

modeling as their common denominator. The first chapter considers a spatial

econometric modeling approach in economic growth and macroeconomic volatility.

The second chapter employs a spatio-temporal econometric modeling technique in

examining house price fluctuations and macroeconomic dynamics, whereas the third

chapter treats a spatio-temporal econometric dynamics of electricity spot prices.

The first chapter, On the link between volatility and growth explicitly treats the

relationship between economic growth and macroeconomic volatility from a spatial

econometrics perspective.1 The influential work of Ramey and Ramey (1995)

highlighted that volatility and growth are negatively related. In contrast, the other

strand of the literature on growth and volatility interactions, e.g. Dejuan and Gurr

1Published in Spatial Economic Analysis, 11: 27-45.

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CONTENTS vii

(2006) finds a positive relationship between growth and volatility. In these model

settings, economies are most of the time considered as independent observations with

no spatial interactions between them. However, evidence in favor of spatial

interactions is now well documented in the literature (LeSage and Fischer, 2008). How

does spatial interaction affect the conventional relationship between macroeconomic

volatility and economic growth? This chapter attempts to answer this question. In

contrast to Ramey and Ramey (1995), we allow cross-country interdependence in

volatility and growth interactions. We use a panel of 78 countries over the period 1970

to 2010. The application of spatial economic analysis to cross-country

volatility-growth relationship is an interesting task as it takes an alternative approach

to demonstrate that globalization and hence tighter links between countries may

influence the conventional relationships between volatility and growth. In order to

capture the spatial interaction between countries, we use a bilateral trade weight

matrix which helps in capturing the appropriate economic interactions between

countries which may not be captured by the standard distance weight matrix. The

classic and the robust Lagrange multiplier (LM) tests applied on the estimated results

of the conventional Ramey-Ramey model indicate non-zero spatial interaction effects

in our data implying the need to account for spatial dependence in growth and

volatility modeling. The estimation results of the unconstrained spatial Durbin model

show that macroeconomic volatility in addition to lowering growth rate of a particular

country, transmits to neighboring countries through trade and lowers neighboring

countries’ growth rate. Growth rates observed in neighboring countries has a positive

effect on growth rate of a particular country. The main results are robust to different

alternative specifications such as adopting geographical distance weight matrix,

country specific and time period fixed effects.

The second chapter, House price fluctuations and macroeconomic dynamics joint

work with Luc Anselin (Arizona State University) is concerned with the space-time

dynamics of house price fluctuations and the macroeconomic dynamics using 373

metropolitan areas in the US from 2001 to 2013. Much of the existing literature on the

interactions between the US housing market and the macro economy present two

major findings. First, house price fluctuations spill over to the macroeconomy over

time (Iacoviello and Neri 2010). Second, house price fluctuations show spatial effects

where price fluctuations from one area transmit to the other areas (Valentini et al.

CONTENTS viii

2013). Using spatial econometric modeling techniques, we combine both the temporal

and spatial effects in examining the interactions between house price movements and

the real economy. We show that house price fluctuations have detrimental effects on

output growth and spillover from one area to another. The loss of output due to house

price fluctuations is more pronounced during the recent financial crisis. Moreover, we

show that house price synchronization has been increasing over time across

metropolitan areas.

The third and final chapter, Space-time modeling of electricity spot prices

co-authored with Niels Haldrup (Aarhus University and CREATES) deals with

space-time econometric modeling of electricity spot prices. Douglas and Popova

(2011) estimate a spatial error model for twelve US spot market regions by allowing

spatial dependence in the disturbance terms of their model. They show that spatial

patterns play a significant role in electricity price dynamics. LeSage and Pace (2009)

argue that a more flexible spatial Durbin model that allows spatial interactions both in

the dependent and independent variables provides better coefficient estimates

compared to a model that allows dependence in the disturbance terms. Using data for

the Nord Pool power market, we derive a space-time Durbin model for electricity spot

prices with both temporal and spatial lags. Joint modeling of both temporal and spatial

adjustment effects is important when prices and loads are determined in a network of

power exchange areas. By using different spatial weight matrices statistical tests show

significant spatial dependence in the spot price dynamics across areas and estimation

of the model shows that the spatial lag variable is as important as the temporal lag

variable in describing the spot price dynamics. We decompose the price impacts into

direct and indirect effects and demonstrate how price effects transmit to neighboring

markets and decline with distance. We conduct a forecasting exercise and we find that

the space-time model has an improved prediction performance for 7 and 30 days

ahead forecasts compared to the non-spatial model. A model with time varying

parameters is estimated for an expanded sample period and it is found that the spatial

correlation within the power grid has increased over time which we interpret as an

indication of an increased degree of market integration within the sample period.

CONTENTS ix

References

Anselin, L. (1988) Spatial Econometrics: Methods and Models. Kluwer Academic

Publishers.

Dejuan, J. & Gurr, S. (2006) On the link between Volatility and Growth: Evidence

from Canadian Provinces. Applied Economics Letters, 11, 279-282.

Douglas, S. & Popova, J. (2011) Econometric Estimation of Spatial Patterns in

Electricity Prices. The Energy Journal, 32, 81-106.

Elhorst, P. (2012) Dynamic Spatial Panels: Models, Methods and Inferences.

Journal of Geographical Systems, 14, 5-28.

Iacoviello, M. & Neri, S. (2010) Housing market spillovers: evidence from anestimated dsge model. American Economic Journal: Macroeconomics, 2, 125-164.

LeSage, J. & Fischer, M. (2008) Spatial Growth Regressions: Model Specification,

Estimation and Interpretation. Spatial Economic Analysis, 3, 275-304.

LeSage, J. & Pace, R. (2009) Introduction to Spatial Econometrics. CRC Press Taylor

and Francis Group, Boca Raton.

Ramey, G. & Ramey, V. (1995) Cross-Country Evidence on the Link between Volatility

and Growth. American Economic Review, 85, 1138- 1151.

x

CONTENTS xi

DANISH SUMMARY

Denne afhandling består af tre selvstændige kapitler om spatial økonometrisk

analyse for tværsnits- og paneldata. Det er veldokumenteret i empiriske og teoretiske

studier, at voksende markedsintegration og globalisering fører til øget afhængighed og

interdependens mellem økonomiske enheder (dvs. geografiske områder, lande, byer,

markedsområder m.v.). Den seneste udvikling i spatial økonometri har givet os et

kraftfuldt værktøj til at undersøge den spatiale dynamik på tværs af økonomiske

enheder. De tre kapitler i denne afhandling er empiriske bidrag til denne litteratur

hvor fællesnæveren er spatial afhængighed. Det første kapitel, On the link between

volatility and growth, omhandler eksplicit sammenhængen mellem økonomisk vækst

og makroøkonomisk volatilitet set ud fra et rumligt økonometrisk perspektiv. Ramey

og Ramey (1995) argumenterer for en negativ sammenhæng mellem volatilitet og

vækst. I modsætning til Ramey og Ramey (1995) tillader vi i dette studie indbyrdes

afhængighed imellem lande og undersøger volatilitet og vækstinteraktioner på tværs af

78 lande for perioden 1970-2010. Estimationsresultaterne for den spatiale model viser,

at makroøkonomisk volatilitet, ud over at sænke væksten i et bestemt land,

transmitteres til nabolande gennem handel og sænker nabolandenes vækstrate.

Det andet kapitel, House price fluctuations and macroeconomic dynamics, er fælles

arbejde med Luc Anselin (Arizona State University) og fokuserer på den

spatio-temporale dynamik for husprisudsving og makroøkonomisk i 373

storbyområder i USA for årene 2001-2013. Ved brug af spatial- økonometriske

modelberegninger viser vi, at husprisudsving har en skadelig afsmittende virkning på

væksten fra et område til et andet.

Det tredje og sidste kapitel, Space-time modeling of electricity spot prices, er skrevet

sammen med Niels Haldrup (Aarhus Universitet, CREATES) og omhandler

spatio-temporal økonometrisk modellering af elspotpriser. Ved hjælp af data for den

nordiske el-børs, Nord Pool, udvikler vi en spatio-temoral Durbin-model for

elspotpriser med både tidsmæssige og spatial dimension. I modsætning til Douglas og

Popova (2011), der estimerer en spatial fejlmodel for tolv amerikanske

spotmarkedsregioner ved at tillade rumlig afhængighed i fejlleddene, udleder vi en

mere fleksibel rum-tid Durbin-model, der tillader spatial afhængighed i både

afhængige og uafhængige variable. Ved at benytte forskellige spatiale vægtmatricer

viser statistiske tests betydelig spatial afhængighed i spotprisdynamikken på tværs af

CONTENTS xii

områder og estimation af modellen viser, at denne afhængighed er lige så vigtig som

den tidsmæssige afhængighed i beskrivelsen af spotprisdynamikken. En model med

tidsvarierende parametre estimeres for en udvidet estimationsperiode og det

konstateres, at den spatiale korrelation inden for elnettet er steget over tid, hvilket vi

tolker som en indikation af en stigende grad af markedsintegration i

estimationsperioden.

Litteratur



Ramey, G. & Ramey, V. (1995) Cross-Country Evidence on the Link between

Volatility and Growth. American Economic Review, 85, 1138- 1151.

CH

AP

TE

R

1ON THE LINK BETWEEN VOLATILITY AND GROWTH: A SPATIAL

ECONOMETRICS APPROACH

Published in Spatial Economic Analysis, 11: 27-45.


Aarhus University and CREATES

Abstract

This paper examines the link between macro volatility and economic growth in the

lens of spatial econometrics. We present an unconstrained spatial Durbin

Ramey-Ramey model. We test the extended model in a panel of 78 countries to

investigate all the possible dimensions along which spatial interactions can affect the

link between macro volatility and growth. In contrast to previous literature, we split

the effects of volatility on growth into direct and indirect effects using partial derivative

impacts approach. We found that both the direct and indirect effects of volatility on

growth are negative; the latter effect suggesting the transmission of volatility shocks to

1

CHAPTER 1. ON THE LINK BETWEEN VOLATILITY AND GROWTH: A SPATIAL ECONOMETRICSAPPROACH 2

neighboring countries. Growth rates observed in neighboring countries has a positive

effect on growth rate of a particular country.

Key words: Spatial effects; volatility; growth; spatial Durbin model

JEL classification: C31; F41; O40

Acknowledgments

I wish to thank two anonymous referees for their invaluable comments and

suggestions. For helpful comments and discussions, I thank Arthur Getis, Bent Jesper

Christensen, Niels Haldrup, Paul Elhorst, and seminar participants of the 61st North

American Regional Council (NARSC) meeting in Washington, D.C., 2014, the 54th

Western Regional Science (WRSA) meeting in Tucson, Arizona, 2015, the 13th

International Workshop on Spatial Econometrics and Statistics in Toulon, France, 2014

and the Danish Graduate Programme in Economics (DGPE) in Fyn, Denmark, 2013.

Any of the errors are solely mine. Financial support from Center for Research in

Econometric Analysis of Time Series - CREATES (DNRF78), funded by the Danish

National Research Foundation is gratefully acknowledged.

1.1 Introduction

How does spatial interaction affect the conventional relationship between macro

volatility and economic growth? This question is particularly plausible in modern

economies where the global economy has moved to closer integration through

cross-border trade and financial flows. Recent theoretical and empirical works

generally focus on the interdependence of economic units and regions implying that

the economy of one country or region is not independent of the economies of others

(Bivand, 1984; Ertur and Koch, 2006; Fingleton and Arbia, 2008). This interdependence

can originate from spatial spillovers stemming from contagion effects or from

unobserved heterogeneity caused by omitted explanatory variables (Ertur and Koch,

2007), foreign knowledge through international trade and foreign direct investment


(Coe and Helpman, 1995), or technology transfer (Barro and Sala-i-Martin, 1997), or

human capital externalities (Lucas, 1988).

The recent surge of interest in spatial modeling has resulted in the application of

spatial econometrics in a wide range of empirical investigations in more traditional

fields of economics, including, among others, growth regressions (Elhorst et al., 2010),

financial markets (Fernandez, 2011), and housing prices (Gerkman, 2010). Particularly,

growth regression in the lens of spatial econometrics has been a growing area of

interest.1

The influential work of Ramey and Ramey (1995) highlighted that volatility and

growth are negatively related. Whereas one direction of the literature (Ramey and

Ramey, 1995; Pindyck, 1991; Bernanke, 1983) documents a negative relationship

between volatility and growth, the other direction of the literature (Dejuan and Gurr,

2006; Dawson and Stephenson, 1997; Koremendi and Meguire, 1985; Grier and

Tullock, 1989) provides a positive link between business cycle fluctuation and growth.

In these frameworks, economies are most of the time considered as independent

observations with no spatial interactions between them. In contrast, evidence in favor

of spatial interactions is now well documented in the empirical literature (LeSage and

Fischer, 2008; Ertur and Koch, 2007; Conley and Ligon, 2002; Moreno and Trehan,

1997). While there is a variety of theoretical reasons and substantial empirical

evidence of interdependence between economies in volatility and growth interactions,

this cross-sectional interdependence has been neglected in the standard literature.

How do macroeconomic volatility and growth interact in the framework of spatial

interactions? This paper attempts to answer this question. The application of spatial

economic analysis to cross-country volatility-growth relationship is an interesting

exercise, as it takes an alternative approach to show that globalization and hence

tighter links between countries, may influence the conventional relationships between

volatility and growth.

In a recent paper, Dewachter et al. (2012) propose a spatial macroeconomic model

for eleven European countries over the period 1981-2008. Using dynamic spatial panel

models, they document that major macroeconomic variables, including inflation,

output gap, and interest rate are interrelated across countries, and a shock that occurs

in a particular country transmits to nearby countries. In another direction of the

1See, for example, Fischer (2011); Elhorst et al. (2010); LeSage and Fischer (2008) and Ertur and Koch (2007).


literature, Falk and Sinabell (2008) apply tools of spatial econometrics in European

regions over the period 1995-2004 and find that spatial interdependence significantly

affects the conventional relationship between volatility and growth. However,

cross-country analysis on the link between volatility and growth from a spatial

perspective is overlooked in the literature. This paper examines the link between

volatility and growth in the lens of spatial econometrics.

Section 1.2 introduces the standard and the spatial Ramey-Ramey model. It is shown

that the econometric specification of the Ramey-Ramey model takes the form of an

unconstrained spatial Durbin model (SDM). Section 1.3 presents the data setup and

the spatial weight matrices used along with the empirical results. In this study, we use a

panel of 78 countries for which we have a complete data set over the period 1970-2010.

We first estimate the standard Ramey-Ramey model as a benchmark, and the results

show a significant negative relationship between volatility and growth. We next allow

spatial interactions and confront the extended models with a panel of 78 countries over

the period 1970-2010.

In order to capture the spatial interaction between countries, we use bilateral trade

weight matrix. The motivation in considering bilateral trade weight matrix instead of

the conventional geographical weight matrix comes from the fact that spatial weight

matrices based on trade intensities are more appropriate in capturing economic

spillovers than the geographical distance weight matrices.2 Countries which trade

more are closer connected economically, e.g. have more correlated business cycles, see

also Frankel and Rose (1998).

The classic and the robust Lagrange multiplier (LM) tests applied on the estimated

results of the conventional Ramey-Ramey model indicate non-zero spatial interaction

effects in our data implying the need to account for spatial dependence in growth and

volatility modeling. We adopt LeSage and Pace’s (2009) partial derivative approach and

decompose the effects of volatility on growth into direct and indirect effects. Our

empirical results indicate some important findings.

First, the spatial autoregressive and spatial error models are rejected in favor of the

unconstrained spatial Durbin model. A number of papers (Elhorst, 2012; LeSage and

Fischer, 2008) argue that the spatial Durbin model produces unbiased coefficient

estimates, also, if the true model is either spatial lag or spatial error model.

2See, for example, Asgarian et al. (2012) and Dewachter et al. (2012) for motivation and an application of bilateral spatial tradeweight matrix.


Second, volatility and growth rates observed in neighboring economies play a

significant role on the link between volatility and growth rate of a particular economy.

We found that growth rates observed in neighboring countries has a positive

significant effect on the growth rate of a particular country, and both the direct and

indirect effects of volatility on growth are negative. The negative indirect effects of

volatility on growth shows that, in addition to depressing a country’s own income

growth, volatility spills over across countries and depresses other countries’ income

growth.

We further examine the robustness of our main results under different model

specifications. One of the potential problems associated with using a spatial weight

matrix constructed from bilateral trade is that volume of bilateral trade and growth

may be determined simultaneously in the long run equilibrium resulting in an

endogeneity problem. We use an inverse distance geographical weight matrix as an

alternative to examine the robustness of the main results to changes in the weight

matrix. The main findings remain the same under the bilateral trade and geographical

weight matrices.

In Section 1.4, we estimate the spatial Ramey-Ramey model by adding country

specific and time period fixed effects. The inclusion of country specific fixed effects

helps in removing possible effects of volatility on growth that may occur because of

differences in growth rates across countries. Similarly, adding time period fixed effects

rules out any possible correlation between volatility and growth over time. We found

significant negative volatility spillover effects on growth after controlling for country

specific and time period fixed effects. Higher growth rate observed in neighboring

countries has a positive effect on growth after controlling both for country specific and

time period fixed effects.

Our approach recognizes that spatial interaction effects exist between countries

that are neighbors to each other and has to be accounted for in growth and volatility

modeling. This is particularly important because nowadays, countries are becoming

more integrated through globalization and trade calling for the need to account for

spatial interactions in growth and volatility models.3 Appropriately determining the

3Kose and Yi (2001) show that an increase in trade and specialization along with a decline in transportation costs induceshigher business-cycle interdependence across countries. Artis and Zhang (1997) examine the effect of the exchange-rate-system(ERM) of the European monetary system on business cycle interdependence across coutries. They show that most countries’business cycle were linked to that of the United States business cycle, but after the formation of the ERM, most countries’ businesscycle shifted to the German business cycle path.


partial derivative effect of changes in volatility on growth is another contribution of

this study. Under our spatial Durbin Ramey-Ramey model, a change in volatility in a

country (say i ) can affect own (direct effect) and neighboring countries’ ( j 6= i ) growth

rate (indirect effect). Explicitly, we isolate the effects of volatility on growth into direct

and indirect (spillover) effects. Section 1.5 presents the conclusion.

1.2 The model

In this section, we extend the Ramey-Ramey volatility and growth model by

allowing spatial effects, which implies economic interdependency across N countries,

i = 1, . . . N .

Our point of departure is the standard Ramey and Ramey (1995) volatility-growth

model specified as

gi t =β0 +β1vi +β2Xi t +εi t , (1.1)

where εi t ∼ N (0, v2i ), g is the annual growth rate of GDP per capita for country i at time

t , vi is the standard deviation of the residuals ε, X represents Levine and Renelt (1992)

variables, namely, initial level of GDP per capita, fraction of investment to GDP, human

capital, and the average growth rate of population. The variance of the residuals, ε, v2

is assumed to vary across countries but constant over time.4

The standard Ramey-Ramey model given in (1.1) ignores possible spatial

interaction effects in analyzing the link between volatility and growth across countries.

The conventional growth regression variables, such as per capita income and

population variables are found to exhibit spatial dependence, implying that

economies can no more be treated as independent in growth regression, see e.g. Ertur

and Koch (2007).5 Income, and hence growth, in a particular country depends on the

income, physical and human capital levels observed in neighboring countries as well.

The full spatial Durbin Ramey-Ramey model takes the form

g = ρW g +π0 +π1v +π2W v +π3X +π4W X +ε, (1.2)

4Note that we allow v2 to vary both across countries and over time later in section 1.4, when we introduce country specific andtime period fixed effects.

5See also LeSage and Fischer (2008) and Elhorst et al. (2010) for a recent SDM specification of the Solow-Swan growth model.


The model in equation (1.2) is known, in the spatial econometrics literature, as the

unconstrained SDM, as it includes the spatially lagged values of both the dependent

and independent variables. It shows that, unlike the classical Ramey-Ramey model,

the link between volatility and growth is not only a function of explanatory variables of

country i itself, but also the growth rate and some other explanatory variables of

neighboring countries.6

First, as in the basic Ramey-Ramey specification, volatility v and a set of Levine and

Renelt (L-R) variables X of a particular country enter the growth equation, see the 3rd

and 5th right hand side terms in equation (1.2). Next, observed values of neighboring

countries growth rate (W g ), volatility (W v), and set of L-R variables (W X ) also enter

the growth equation of a particular country. This is captured, respectively, by the 1st,

4th, and 6th right hand side terms in equation (1.2). The parameter ρ quantifies the

impact of growth rate of nearby countries on the growth rate of a particular country i .

Under the assumption of no spatial interactions, i.e ρ = 0, π2 = 0 and π4 = 0, equation

(1.2) produces the conventional Ramey-Ramey model given in (1.1).

Economic theory suggests that in an open economy, the level of income, and hence

the growth rate of the domestic country is a positive function of the level of income

and hence the growth rate of a trading partner or neighboring country (e.g. Blanchard,

2013). Consider, for example, an increase in the income of a trading partner country.

The increase in income of the foreign (trading partner) country leads to an increase in

net export of the domestic economy which in turn increases the domestic income and

hence growth. Observed values of neighboring countries growth rate (W g ) in equation

(1.2) captures this potential relation.

The term W v implies that business cycle fluctuation observed in nearby economies

might play a role in the growth rate of the particular economy. Empirical studies (e.g.

Canova and Dellas, 1993) show that business cycle fluctuation in a particular country

propagates to other countries through international trade implying that macro

volatility observed in nearby countries will have an important effect on the growth rate

of a particular country. Similarly, the matrix W X contains linear combinations of a set

of control variables such as initial per capita income, human capital, investment ratio,

and population growth in nearby countries. This term captures the hypothesis that an

observed initial per capita income, human capital, investment ratio, and population in

6Neighbor here refers to economic neighborhood, not necessarily the mere geographical closeness.


nearby economies play a role in the growth rate of a particular economy.

Moreover, the SDM in (1.2) nests various spatial models as a special case. Setting the

restriction that π2 = 0 and π4 = 0 yields a spatial autoregressive (SAR) model specified

as

g = ρW g +π0 +π1v +π3X +ε. (1.3)

Imposing the non-linear restrictions that π2 = −ρπ1 and π4 = −ρπ3 produces the

spatial error model (SER) of the form

g =π′0 +π1v +π3X + (I −ρW )−1ε, (1.4)

where π′0 = (I −ρW )−1π0. In the spatial econometrics literature, the SDM is preferred

over the SAR and the SER models, see Elhorst (2012). This is because the SDM produces

unbiased coefficient estimates, also if the true data generating process is either spatial

lag or spatial error model.

1.3 The empirics

1.3.1 Data

We study the relationship between growth and volatility using a large data set drawn

from different sources. Our sample consists of 78 countries for which we have complete

data for the period 1970-2010. Following the literature on growth empirics (Ramey and

Ramey (1995)), investment, real income, population, and government spending data

are drawn from the latest version of Penn World Table (Heston et al., 2012). Bilateral

trade (import and export value) data is interpolated from the International Monetary

Fund (IMF) financial statistics. The human capital data is extracted from Barro and Lee

(2010) as in Ramey and Ramey (1995), see Appendix for details of data sources and list

of countries used.

In our benchmark setup, the dependent variable is the annual growth rate of GDP

per capita (GYP), the explanatory variables include volatility measured as a standard

deviation of the residuals of the growth equation (VOL), the 1970 initial level of GDP per

capita (GDPO), fraction of investment to GDP (INV), human capital (HUC) measured

as the average years of schooling for individuals over age 25 as in Ramey and Ramey

(1995), and the annual growth rate of the population (POP) over the period 1970-2010.


1.3.2 Non-spatial Ramey-Ramey model

To begin with, we determine the mean and standard deviation of GDP per capita

annual growth rates over time for each country and investigate the cross-sectional

relationship between volatility and growth. The regression result of mean growth (4yi )

on the standard deviation of growth (vi ) over the period 1970-2010 gives

4yi = 0.0157−0.0186vi

(0.00) (0.00),

p values in parentheses.7 This regression result indicates a statistically significant

negative relationship between volatility and growth in the simple cross-sectional

specification. Specifically, the estimated coefficient on volatility(vi ) indicates that a

higher standard deviation of GDP per capita is associated with a lower growth rate.

This result is similar to Ramey and Ramey (1995) but contradicts Dejuan and Gurr

(2006).

1.3.3 The spatial Ramey-Ramey model

We now examine the relationship between volatility and growth taking spatial

interaction effects into account. For this end, we estimate the spatially augmented

models using quasi maximum likelihood. Before we investigate the spatial results, we

first discuss the spatial weight matrices used in the current study and potential

channels of spatial interdependence.

In the spatial econometrics literature, there is little guidance in the choice of the

correct spatial weights in an empirical application. The usual tradition in constructing

the spatial weight matrix has been geographical distance. However, it is not obvious that

geography is the most relevant factor in economic interdependence between countries

(Case et al., 1993). This is because, geographical distance may not account for the basic

role of trading partners of a country.

7Note that volatility is measured as the standard deviation of growth in the simple cross-sectional specification. In the paneldata framework as in equations (1.1) and (1.2), volatility is measured as the standard deviation of the residuals of the growthequation. See Dejuan and Gurr (2006), Dawson and Stephenson (1997) and Ramey and Ramey (1995) for details. Whereas themeasure of volatility as the standard deviation of growth is sometimes called the unconditional volatility of growth, the measure ofvolatility as the standard deviation of the residuals of the growth equation is called conditional volatility of growth. Unless otherwisestated, volatility in this paper refers to the latter definition.


In this study, we use bilateral trade weight matrix to capture the relative closeness

of countries to one another. The use of bilateral trade weight matrix as a measure of

economic linkages among countries is based on the economic theory, which suggests

that the existence of cross-border trade supports the prediction that economic

outcomes across nations is not independent (Canova and Dellas, 1993; Garcia-Vega

and Herce, 2002). This better captures the economic interdependency among

countries unlike the usual geographic distance weight matrix used in the spatial

econometrics literature. Large value of trade between two countries implies higher

dependence between the countries and increase the degree economic interactions.

For any pair of countries i and j , i 6= j , we define the general terms of bilateral trade

weight matrix WT as

wi j =Xi j + I M j i∑k=N

k=1 Xi k +∑k=N

k=1 ki,

where Xi j is the value of export of goods and services from country i to j , and I M j i is

the value of import of goods and services from country j to country i during the period

1970-2010. Once WT has been computed, each of its row is divided by the sum of its

corresponding elements so that the row sums to unity. Such specification of the weight

matrix indicates that the higher the share of exports and/or imports of country i from

country j , the more economically interdependent the countries i and j are resulting in

higher shock spillovers from one country to the other. Our choice of constructing the

weight matrix is similar to that of Case et al. (1993) in that we rely on economic weight

matrix instead of the geographical weight matrix.

1.3.4 Results and discussion

In this section, we analyze the spatial Durbin Ramey-Ramey model given in equation

(1.2) on a panel of 78 countries over the period 1970-2010. We begin our analysis by

estimating the non-spatial Ramey-Ramey model, i.e, a model with the restrictions ρ =0, π2 = 0 and π4 = 0 in (1.2).


Table 1.1: Basic and Spatially augmented Ramey-Ramey models

Dependent variable: Growth rate in per capita GDP

Independent variable (1) (2)

Constant -0.007 (0.003)***

VOL -0.019 (0.005)***

INV 0.666 (0.045)***

HUC 0.435 (0.042)***

GDPO -0.003 (0.000)***

POP 0.0007 (0.000)***

Direc effect VOL -0.270 (0.012)***

Indirect effect W*VOL -0.117 (0.074)*

Total effect VOL -0.387 (0.079)*

Direct effect INV -0.00002 (0.015)

Indirect effect W*INV -0.071 (0.071)

Total effect INV -0.071 (0.081)

Direct effect HUC -0.034 (0.026)

Indirect effect W*HUC 0.180 (0.102)*

Total effect HUC 0.147 (0.113)

Direct effect GDPO 0.0001 (0.001)

Indirect effect W*GDPO -0.019 (0.009)***

Total effect GDPO -0.019 (0.009)**

Direct effect POP -0.001 (0.000)***

Indirect W*POP 0.002 (0.002)

Total effect POP 0.001 (0.002)

ρ 0.679 (0.015)***

LM test: no spatial lag 989.82 (0.000)***

Robust LM test: no spatial lag 66.99 (0.000)***

LM test: no spatial error 1485.29 (0.000)***

Robust LM test: no spatial error 562.48 (0.000)***

Linktest 1.49 (0.137)

Mean VIF 1.43

Ramsey Test 54.65 (0.000)***

Wald test lag 9.64 (0.000)***

Wald test error 4.30 (0.000)***

N 3198 3198

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in parenthesis

for model results and p-values are in parenthesis for LM and Wald test results.


Anselin et al. (1996) provide LM test for spatial interaction effects among the

dependent variable and LM test for spatial interaction among the error terms. The

former test is known as LM lag model, and the latter test is known as the LM error. The

robust LM lag tests spatial interaction effects among the dependent variable in the

presence of error autocorrelation. Similarly, the robust LM error test tests spatial

interaction effects among the error terms in the presence of a spatially lagged

dependent variable. Both the classic and the robust LM tests are based on the OLS

residuals of the non-spatial model.8 The first column of Table 1.1 reports the two-step

maximum likelihood estimation results of the basic Ramey-Ramey model.9 As shown

in the table, the restricted (non-spatial model) is rejected based on the classic and

robust LM tests. The simple Ramsey test for omitted variable, for example, also rejects

the null hypothesis that there is no omitted variable under the standard Ramey-Ramey

specification. We thus proceed to the spatial model estimation.

1.3.5 Direct and indirect effects

In the SDM that includes the spatial lags of both the dependent and independent

variables, a change in a single explanatory variable in country i has a direct effect on

country i itself as well as an indirect effect on other countries j 6= i .

Consider the SDM given in equation (1.2) as a point of departure

g = (I −ρW )−1(π0 +π1v +π2W v +π3X +π4W X +ε).

The matrix of partial derivatives of g with respect to the explanatory variable v (for

example) for i = 1, ..., N gives

[∂g∂v1

. . . ∂g∂vN

]= (I −ρW )−1

π1 w12π2 . . . w1nπ2

w21π2 π1 . . . w2nπ2

. . . . . .

. . . . . .

. . . . . .

wn1π2 wn2π2 . . . π1

,

where wi j is the (i , j )th element of the weight matrix W . LeSage and Pace (2009) show

8For technical details on LM tests, see Anselin et al. (1996).9See Appendix B for general procedures and Greene (2011) for proofs and details on the two-step maximum likelihood

estimation.


that the direct effect is measured by the average of the diagonal elements, while the

indirect or spatial spillover effect is measured by the average of either the row sums

or the column sums of the non-diagonal elements. However, because the numerical

magnitudes of either the row sums or the column sums is the same, it does not matter

which one is used for calculating the magnitudes of the indirect effects.10

Gibbsons and Overman (2012), however, argue that the parameter of the spatially

lagged dependent variable might pick up the effects of omitted spatially lagged

variables resulting in biased indirect effects. However, LeSage and Pace (2009) strongly

suggest a partial derivatives impact approach, because the standard point estimates of

spatial regression model specifications may lead to erroneous conclusions. Further in

this paper, we found the direct and indirect effects estimation interesting, because it

enables to isolate the impacts of volatility on growth into direct and indirect effects. We

thus use direct and indirect effects estimation technique following LeSage and Pace

(2009).

The spatial panel quasi maximum likelihood estimation results of the spatially

augmented Ramey-Ramey model are reported in column (2) of Table 1.1. We estimate

our spatial Durbin Ramey-Ramey model in two steps: 1) Estimate equation (1.2) using

spatial quasi maximum likelihood setting π1 = 0 and π2 = 0 to obtain the standard

deviation of residuals. 2) The estimated standard deviations from step (1) were then

included as variables in the main equation, and the model was re-estimated using

quasi maximum likelihood.11 The spatial Ramey-Ramey model produces some

important results.

First, the growth rate of neighboring countries has a positive and statistically

significant effect on the growth rate of a particular country. This is inline with many

empirical results (LeSage and Fischer, 2008; Ertur and Koch, 2007) that provide

considerable support to the theory that the growth rate of neighboring countries

positively affects the growth rate of a particular country. This implies that observed

growth rates in neighboring countries play an important role on the link between

volatility and growth of a particular country.

Second, the coefficient of volatility on growth (direct effect) is negative and

10LeSage and Pace (2009) suggest simulating the distribution of the direct and indirect effects using the variance-covariancematrix implied by the maximum likelihood estimates to make inferences about the statistical significance of the direct and indirecteffects.

11Note that Ramey and Ramey (1995) use similar procedure (at least partially) in their model, see their footnotes associated withtable 4. As indicated in footnote 9, Greene (2011) presents detailed proofs and motivations for adopting the two step estimationprocedure.


statistically significant with a coefficient estimate of -0.270 at 1% level of significance.

This shows that macroeconomic volatility and growth are negatively correlated under

the spatially extended Ramey-Ramey model. This is consistent with the findings in

Ramey and Ramey (1995), Pindyck (1991), and Bernanke (1983). One of the potential

explanations for a negative relationship between output volatility and growth is

irreversibility in investment. Theoretical analysis suggests that if there are

irreversibilities in investment, then increased volatility can lead to lower investment

and hence lower growth (Aghion et al., 2010).

Third, the indirect effect of volatility on growth is negative and significant with a

coefficient estimate of -0.117 at 10% level of significance. This effect shows that

volatility transmits to neighboring countries and hampers growth. The total effect of

volatility (the sum of the direct and indirect effects) is significant with a coefficient

estimate of -3.87 at 10% level of significance. The recent financial crisis, that

propagated from the United States to the rest of the world, might reflect our finding.

Bacchetta and Wincoop (2013), for example, show that synchronization of business

cycle panic across countries through trade has resulted in the diffusion of the financial

crisis from the United States to the rest of the world during the period 2008-2009.

One can perform Wald tests to examine whether the SDM estimated in column (2)

reduces either to the spatial lag or the spatial error model. Both Wald test lag and Wald

test error reject the null hypothesis that the SDM reduces either to the spatial lag or

spatial error model.

One of the potential concerns of the spatial Durbin model estimation results so far

is the endogeneity of the bilateral trade weight matrix with our dependent (growth)

variable, see e.g. Wacziarg (2001). In order to examine the sensitivity of our main results

to the changes in the weight matrix, we use an inverse distance spatial weight matrix.

The inverse distance weight matrix is specified as

W1 = w1i j∑j w1i j

, where w1i j =0 i f i = j

d−1i j other wi se,

where di j is the great-circle distance between country capitals.12

12The great-circle distance, the shortest distance between any two points, is computed as:di j = r adi ous x cos−1[cos | long tui dei − l ong tude j | cosl ati tudei cosl ati tude j + si nl ati tudei si nl ati tude j ]


Table 1.2: Spatially augmented Ramey-Ramey models

Dependent variable: Growth rate in per capita GDP

Independent variable (1)

Direct effect VOL -0.012 (0.006)**

Indirect effect W*VOL -0.790 (0.454)*

Total effect VOL -0.802 (0.459)*

Direct effect INV 0.013 (0.015)

Indirect effect W*INV 0.607 (0.470

Total effect INV 0.610 (0.479)

Direct effect HUC -0.036 (0.024)

Indirect effect W*HUC 2.414 (0.723)***

Total effect HUC 2.378 (0.732)***

Direct effect GDPO -0.011 (0.009)

Indirect effect W*GDPO -0.902 (0.654)

Total effect GDPO -0.913 (0.663)

Direct effect POP -0.002 (0.001)**

Indirect W*POP -0.029 (0.010)**

Total effect POP -0.031 (0.010)**

ρ 0.879 (0.012)***

Wald test lag 8.68 (0.000)***

Wald test error 4.10 (0.000)***

N 3198

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level.

Standard errors in parenthesis for coefficient estimates.

p-values are in parenthesis for Wald tests. Inverse distance

weight is used in the estimation of the SDM.

The estimation results of equation (1.2) under the geographical weight matrix are

reported in column (1) of Table 1.2. The results show that the main findings are similar

to the results obtained using bilateral trade weight matrix. Where as both the direct

and indirect effects of volatility on growth remain negative and significant, spatially

lagged growth rate has a positive significant effect on the growth rate of a particular

country. The indirect effect of volatility takes a higher (in absolute terms) value than the

direct effect under the geographical weight matrix. One reason might be the fact that


the geographical distance weight matrix may not appropriately capture the potential

economic interdependence between countries.

1.4 Alternative regression frameworks

In this section we re-examine our previous main results by including

country-specific and time period fixed effects. For this end, we respecify volatility in

such a way that it varies both across countries and over time. Whereas the inclusion of

country specific fixed effects helps in removing possible effects of volatility on growth

that may occur because of differences in growth rates across countries, adding time

period fixed effects rules out any possible correlation between volatility and growth

over time.

In order to estimate the spatial Durbin Ramey-Ramey model under

country-specific and time period fixed effects specification, we first identify a variable

that affects output volatility both across countries and over time. Ramey and Ramey

(1995) identify government spending as a source of volatility13 and estimate system of

equations of the form

gi t =β0 +β1vi t +β2Xi t +εi t . (1.5)

v2i t =α0 +α1µ

2i t , (1.6)

where vi t is the standard deviation of the residuals εi t , µ2i t is the square of the

estimated residual for country i in period t from a government spending growth

equation that contains a constant term, two lags of GDP, two lags of the log levels of

government spending, a linear and quadratic trend, and country dummies for each

country. Ramey-Ramey then investigate whether the variances of the innovations in

the growth equation are related to the squared residuals of the government spending

growth equation; if they are related, we have a measure of volatility that varies both

across countries and over time. This is important to investigate if volatility and growth

are negatively related after controlling for country-specific and time period fixed

effects.

13There could be different sources of volatility, taxes, for example, see Posch and Wälde (2011).


Consider government spending as a source of volatility and the corresponding

spatial Durbin model of the specifications in (1.5) and (1.6) above is

g = ρW g +π0 +π1v +π2W v +π3X +π4W X +ε. (1.7)

v2 =α0 +α1µ2. (1.8)

We proceed in two steps: 1) We estimate country specific government spending

growth equation that contains an intercept, two lags of government spending (in log

terms), two lags of GDP per capita (in log terms), two lags of GDP per capia (in log

terms) from neighboring countries, a linear and quadratic time trend and a dummy

variable for post 2006. 2) We investigate whether the variances of the innovations in

the growth (main) equation are related to the squared forecast residuals of the

government spending growth equation. As stated earlier, if they are significantly

related, then we have a measure of volatility that varies both across countries and

time, and we can then investigate growth-volatility relationships taking into account

country-specific and time period effects.

The parameters of interest in equation (1.7) and (1.8) are π1, π2, ρ, and α1. Whereas

π1 relates the effect of own volatility (direct effect) on output growth, π2 relates the

effect of neighboring countries’ volatility (indirect effect) on output growth. The

relationship between the squared innovations to government spending and the

variances of output growth are captured by the parameter α1. The parameter ρ relates

the effects of neighboring countries’ growth rate on output growth.


Tab

le1.

3:Sp

atia

llyau

gmen

ted

Ram

ey-R

amey

mo

del

s,fi

xed

effe

cts

Dep

end

entv

aria

ble

:Gro

wth

rate

inp

erca

pit

aG

DP

Ind

epen

den

tvar

iab

le(1

)(2

)(3

)(4

)(5

)

Dir

ecte

ffec

tVO

L-0

.140

(0.0

08)*

*-0

.161

(0.0

12)

-0.0

17(0

.012

)-0

.150

(0.0

12)*

-0.1

43(0

.009

)*

Ind

irec

teff

ectW

*VO

L-0

.066

(0.0

35)*

-0.0

74(0

.042

)*-0

.086

(0.0

48)*

-0.1

05(0

.043

)**

-0.1

07(0

.046

)**

Tota

leff

ectV

OL

-0.2

06(0

.058

)**

-0.2

35(0

.045

)**

-0.1

03(0

.049

)**

-0.2

55(0

.046

)**

-0.2

0(0

.052

)**

ρ0.

268

(0.0

52)*

**0.

249

(0.0

37)*

**0.

257

(0.0

37)*

**0.

274

(0.0

37)*

**0.

272

(0.1

09)*

**

α1

0.01

6(0

.003

)***

0.01

8(0

.004

)***

0.01

5(0

.003

)***

0.01

8(0

.002

)***

0.01

6(0

.002

)***

Co

un

try

fixe

def

fect

sn

oye

sn

oye

sye

s

Tim

efi

xed

effe

cts

no

no

yes

yes

yes

Gro

wth

ofg

over

nm

ent

no

no

no

no

yes

Wal

dte

stla

g1.

92(0

.074

)*41

4.60

(0.0

00)*

**21

1.73

(0.0

00)*

**41

6.02

(0.0

39)*

**3.

23(0

.039

)**

Wal

dte

ster

ror

2.94

(0.0

12)*

*33

0.67

(0.0

00)*

**0.

23(0

.924

)41

1.56

(0.0

45)*

**3.

10(0

.045

)***

N31

9831

9831

9831

9831

98

No

tes:

***

(**,

*)d

eno

tes

sign

ifica

nce

at1%

(5%

,10%

)le

vel.

Stan

dar

der

rors

are

inp

aren

thes

isfo

rm

od

elre

sult

san

dp

-val

ues

are

inp

aren

thes

isfo

rW

ald

test

resu

lts.

We

incl

ud

edth

eL-

Rva

riab

les

wh

enco

un

try

fixe

def

fect

sar

eex

clu

ded

,an

dth

etr

end

isin

clu

ded

wh

enti

me

fixe

def

fect

sar

eex

clu

ded

.


The results are reported in Table 1.3. The first column shows the results, when all

the L-R variables (both own and neighbors), two lags of GDP per capita (both own and

neighbors), and trend are included. We do not report the coefficient estimates on

controls to conserve space. The estimates of α1 suggest that the variances of the

growth innovation are significantly related to the squared innovations of the

government spending. The coefficient estimate of π1 shows that volatility has a

significant negative effect on growth. Similarly, the indirect effect of volatility on

growth captured by the coefficient estimate of π2 also suggests that volatility observed

in nearby countries has a significant negative effect on growth at 10% level of

significance.

The second column reports the results of the model when country-specific fixed

effects are included. The country- specific fixed effects specification removes any

effect of volatility on output growth that may occur because of growth rate differences

across countries. The coefficient estimate of π2(indirect effect) shows that volatility

that spills over from neighboring countries has a negative and significant effect on

growth. Observed business cycle fluctuation from neighboring countries’ affects the

domestic growth rate negatively. Own volatility has only negative partial correlation

with growth. Ramey and Ramey also found negative but insignificant relationship

between volatility and growth after controlling for country specific fixed effects. But

the total effect of volatility on growth remains significantly negative with a 5% level of

significance.

The third column shows the results when time period (year dummies) fixed effects

are included but not country- fixed effects. Time period fixed effects rule out possible

correlation between volatility and growth over time. The control variables include all

the L-R (both own and neighbors) variables and two lags of GDP per capita (both own

and neighbors) excluding initial per capita GDP. The indirect effect of volatility on

growth is negative and significant (at 10% level), but the direct effect becomes

insignificant.

In the fourth column, we include both country-specific and time period fixed

effects. This specification removes any effect of volatility on growth that may arise due

to variation in growth rates across countries and over time. The estimation shows that

both the direct and indirect effects of volatility on growth is significantly negative.

The last column of Table 1.3 shows estimation results of the model with government


spending as an additional control variable along with both country-specific and time

period fixed effects. The motivation of including government spending is that in case

the measure of government spending volatility is capturing some effects on growth. The

coefficient estimates are similar to the previous specifications except slight changes in

the coefficient estimates.

In general, the direction of the relationship between volatility and growth remains

negative across all specifications. The spatially lagged growth rate, on the other hand,

has positively significant effect on growth in all specifications. Further, the Wald tests

suggest that both the SAR and SER models are rejected in favor of the SDM across all

specifications except in column (3), where the SDM is rejected in favor of the SER

model where time period fixed effects are included. However, estimating a general

SDM produces unbiased coefficient estimates, also if the true data generating process

is either spatial lag or spatial error model.

This result is a new direction in investigating the link between volatility and growth.

Previous papers focus on addition of some variables to investigate the link between

volatility and growth. Our results indicate that observed growth rate and volatility in

neighboring countries are important factors on the link between growth and volatility

of a particular country. Fingleton (2007) notes that in the present globally

interdependent economic system, events, decisions, and actions made in one country

may have important effect for many other countries, implying that countries can no

more be treated as independent units in many economic processes.

1.5 Conclusion

The spatial econometrics literature points out that spatial interactions in many

economic processes affect the conventional relationship of variables. We investigate

the link between growth and volatility allowing for spatial interactions between

countries. We spatially augment the Ramey-Ramey model and show that global factors

are important in investigating the relationship between growth and volatility.

We empirically test the extended models across a sample of 78 countries over the

period 1970-2010 for which we have a complete data set. First, we estimate the basic

Ramey-Ramey model. The relationship between volatility and growth is negative in all

the estimation results under the non-spatial model. The classic and the robust


Lagrange multiplier (LM) tests performed on the estimated results of the conventional

Ramey-Ramey model indicate significant spatial interaction effects in our data,

implying the need to account for spatial dependence in growth and volatility

modeling. Accordingly, the spatially extended model is estimated allowing possible

spatial interactions between countries. In order to capture the spatial interactions

between countries, we use bilateral trade weight matrix.

We consider both the direct and indirect effects of volatility on growth across

countries. We found that the direct effect of volatility on growth is negative. The

negative indirect effect of volatility on growth also shows that volatility propagates to

other countries and hence depresses economic growth. Higher observed growth rates

in nearby countries, on the other hand, improves the growth rate of a particular

country.

The results show that the negative effect of volatility on growth mainly comes from

the volatility of innovations to the income growth. Moreover, we also examine the

relationship between growth and volatility in a model, where the variance of

innovation to output growth is related to the variance of innovations to government

spending. We found a significant negative spillover effect of volatility to other

countries even after controlling for country-specific and time period fixed effects.

The theoretical extension and the empirical finding obtained in this paper is a new

direction in investigating the link between volatility and growth. Previous papers

mainly focus on addition of some variables in the standard Ramey-Ramey model

neglecting possible spatial interactions between countries. Our results indicate that

observed growth rate and volatility in neighboring countries are determinant factors

on the interactions between volatility and growth of a particular country.

This analysis implies that controls for neighboring countries’ growth rates and

volatility should be included in the conventional growth-volatility regressions. This

paper opens up interesting future research avenues. One can derive a general spatial

Ramey-Ramey model from a theoretical growth model along the lines of Ertur and

Koch (2007). Investigating the relationship between growth and volatility on a dynamic

stochastic equilibrium setup from a spatial econometrics perspective is another

possible future area of research.


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1.7 Appendix

Appendix A. List of countries and data sources used

Table 1.4: List of 78 countries in the sample during the period 1970-2010

Country Code Country Code

Afghanistan AFG Malta MLTAlgeria DZA Mauritius MUSArgentina ARG Mexico MEXAustria AUS Mozambique MOZAustralia AUT Netherlands NLDBangladesh BGD Nepal NPLBarbados BRB New Zealand NZLBelgium BEL Nicaragua NICBolvia BOL Niger NERBrazil BRA Norway NORCanada CAN Pakistan PAKChile CHL Panama PANColombia COL Paraguay PARCosta Rica CRI Papua New Guinea PNGCyprus CYP Peru PERDenmark DNK Philippines PHLDominican Republic DOM Portugal PRTEcuador ECU Senegal SENEl Salvador SLV Sierra Leone SLEFiji FJI Singapore SGPFinland FIN South Africa ZAFFrance FRA Spain ESPGermany GER Sri Lanka LKAGhana GHA Sudan SDNGreece GRC Switzerland CHEGuatemala GTM Syria SYRGuyana GUY Thailand THAHaiti HTI Trinidad and Tobago TTOHunduras HND Togo TGOIceland ISL Turkey TURIndia IND Tunisia TUNIran IRN Uganda UGAIraq IRQ United Kingdom GBRItaly ITA United States USAJamaica JAM Uraguay URYJapan JPN Venezuela VENJordan JOR Zambia ZMBKenya KEN Zimbabwe ZWELiberia LBRMalawi MWI


Table 1.5: Data sources

Variable name Source

Real GDP per capita Penn World Tables (2012)

https://pwt.sas.upenn.edu/

Share of investment to GDP Penn World Tables (2012)


Human capital Barro and Lee (2010)

http://rbarro.com/data-sets/

Population Penn World Tables (2012)


Import and export IMF Financial Statistics

http://www.imf.org/external/data.htm

Government spending Penn World Tables (2012)


Appendix B. Two-step maximum likelihood estimation procedure

We present the two-step estimation procedures from Greene (2011). Suppose we

have two models, model (1.1) and model (1.2) with distributions, respectively,

f1(y1 | x1, θ1) and f2(y2 | x2θ1θ2) where the first model appears in the second but not

the reverse. Estimation procedure in two steps proceeds:

1. Estimate θ1 by maximum likelihood

2. Estimate θ2 by maximum likelihood in model (2) with θ1 obtained from step 1.

The theoretical support for the consistency of θ2 is essentially that if θ1 were known,

then the results would hold true for estimation of θ2, and because asymptotically

pl i mi t θ1 = θ1. For detailed proofs and more arguments in favor of the two step

estimation procedure, see Greene (2011).

Given the standard growth-volatility model of the form

gi t =β0 +β1vi +β2Xi t +εi t (i )

1. We estimate model (i) using quasi maximum likelihood assuming β1 = 0 to obtain

the standard deviation of the residuals. Our model with β1 = 0 is equivalent to model

(1) in Greene (2011).

2. The estimated standard deviations from step 1 were then included as variables in


the main equation and the model was re-estimated using quasi maximum likelihood.

The main equation where β1 6= 0 is equivalent to model (1.2) in Greene (2011). The

model estimated in step 1 appears in the model estimated in step 2 but not the reverse.

One advantage of the two-step estimation procedure over the joint maximum

likelihood is that if either model is misspecified, then the joint estimates of both

models will be inconsistent which is not the case in the two-step estimation

procedure. The other advantage of the two-step estimation procedure is that

maximizing the joint log likelihood may be numerically complicated, see Greene

(2011) for details.

CH

AP

TE

R

2HOUSE PRICE FLUCTUATIONS AND MACROECONOMIC

DYNAMICS



Luc Anselin

Arizona State University

Abstract

This paper investigates the impact of house price movements on output in a

space-time dynamic framework. The transmission of house price fluctuations to the

macroeconomy both across space and over time is explicitly considered through

spatial econometric modeling techniques. Using 373 metropolitan areas in the US

from 2001 to 2013, it is shown that house price fluctuations have detrimental effect on

output growth and spillover from one location to another. The loss of output due to

house price fluctuations is more pronounced during the recent financial crisis. The

time varying recursive estimation of the space-time econometric model shows that the

31

CHAPTER 2. HOUSE PRICE FLUCTUATIONS AND MACROECONOMIC DYNAMICS 32

coefficient of spatial correlation has been increasing over time, reflecting an increasing

trend in house price synchronization.

Keywords: House price fluctuations; output growth; space-time modeling

JEL classification: E30; E32

Acknowledgments

We would like to thank seminar participants of Department of Economics, Arizona

State University, the 62nd North American Regional Science Council (NARSC) meeting

in Portland, Oregon and the Danish Graduate Programme in Economics (DGPE)

workshop in Sandbjerg, Denmark for their helpful comments and discussions. An

earlier version of this paper was circulated under the title “House price fluctuations

and the business cycle dynamics”. Girum Dagnachew Abate gratefully acknowledges

financial support from Center for Research in Econometric Analysis of Time Series -

CREATES (DNRF78), funded by the Danish National Research Foundation.

2.1 Introduction

The recent financial crisis caused by the US housing market crash has led many

researchers in the field to consider the housing sector as a source of macroeconomic

fluctuations, see, for example, Cesa-Bianchi (2013), Iacoviello and Neri (2010), and Liu

et al. (2013). Many of the existing studies on the interactions between the US housing

market and the macroeconomy present two important findings. First, house price

fluctuations spill over to the macroeconomy over time (Holly et al. 2010 and Iacoviello

and Neri 2010). Second, house price fluctuations show spatial effects where price

fluctuations from one location transmit to the other locations (Kuethe and Pede 2011

and Valentini et al. 2013).1 Motivated by this evidence, two interesting questions arise.

(1) How big are the spillovers from the housing market to the real economy? And (2)

what is the nature of housing market spillover from one location to the others?

This paper investigates the impact of house price fluctuations on the

macroeconomy in a joint space-time dynamic framework. The transmission of house

1Location in this particular context refers to any economic unit, e.g. country, region, ZIP code or metropolitan city.


price fluctuations to the real economy both across location (space) and over time is

explicitly considered through spatial econometric modeling techniques. Recent

advances in spatial econometrics provide very interesting and powerful tools for

examining the linkages between the housing market and the real economy both across

space and over time. Because fluctuations in house prices affect the wider economy,

proper understanding of the interactions between house price fluctuations and the

real economy is very important for economic stabilization policies.

In equilibrium models of the housing market and the macroeconomy (see e.g.

Cesa-Bianchi 2013; Iacoviello and Neri 2010; Iacoviello 2005 and Monacelli 2009)

house price changes affect macroeconomic aggregates through the collateral

constraint. Given financial market imperfections, changes in house prices affect

household’s wealth and the capacity of borrowing, investment and consumption.

Specifically, an increase in house price improves the household’s wealth status and

enhances borrowing capacity, investment, and consumption. A boom and subsequent

downturn in the housing market amplifies cyclical fluctuations in the real economy.

Theoretical works by Bernanke et al. (1999) also stress the important linkages between

asset prices (house prices) and the real economy. Similarly, Liu et al. (2013) argue that

housing market shocks are important sources of macroeconomic fluctuations.

A strand of empirical studies (see Hirata et al. 2013; Leamer 2007 and Bordo and

Jeanne 2002) show that house prices exhibit frequent boom and bust and such

housing busts can be very costly in terms of output loss. Figure 2.1 plots the standard

deviation of house prices and output growth for a randomly selected samples of 28 US

metropolitan statistical areas during 2001-2013. The graph shows that a high

fluctuation in house prices is associated with a lower output growth rate during the

sample period. This empirical evidence is supported by earlier studies. Leamer (2007),

for example, shows that there are strong linkages between movements in the housing

markets and business cycles in the US. Stephens (2012) also argues that fluctuations in

house prices hurt the wider economy in different ways. During boom period, there is a

temptation for individuals to overextend borrowing. House price volatility also creates

risk of unsustainable house price for lenders. Moreover, an increase in house price

volatility increases the probability of negative home equity, and mortgage foreclosure

losses become worse.2

2See Miller and Peng (2006) and Penning-Cross (2013) for further discussions.


Figure 2.1: Plot of output growth and volatility of house prices for a sample of 28 MSAs

- - -Standard deviation of prices —Output growth


Many theoretical and empirical works also show that housing markets are

characterized by spatial patterns. Can (1992) states that the value of a house at a

particular location is dependent on the value of houses at nearby locations. Buyers and

sellers, for example, may use similar sale prices in a neighborhood as references for a

transaction sales price, see Anselin (2003). This indicates that the price of a particular

house will affect the price of neighboring houses, indicating that appropriate modeling

of the interactions between the housing market and macroeconomic fluctuations calls

for both the spatial and temporal dynamics. Meen (1999) also states that a

perturbation in house prices in a given location spills over to other locations, leading

to a global effect on house prices in all other locations. Anselin and Lozano-Gracia

(2009) argue that spatial patterns in the housing market could arise from a

combination of spatial heterogeneity and spatial dependence.3 For example, spatial

heterogeneity may result from spatially differentiated characteristics of demand,

supply, and institutional barriers. In a cross-country framework, Cesa-Bianchi (2013)

and others document that movements in house prices are highly synchronized across

countries and house price fluctuations transmit from one country to the other

through, for example, trade and interest rates. Holly et al. (2010) and Baltagi and Li

(2014) also document that US housing markets show significant spatial effects.

While much of the existing research on the interactions between house prices and

the real economy focuses on the temporal dynamics, the links between house prices

and the real economy in a space-time setup have been less thoroughly researched. This

paper aims to fill part of this gap.

We use rich house price data sets across 373 US metropolitan statistical areas

(MSAs) during the period 2001 to 2013. The disaggregated panel data at MSA level

feature some important advantages over aggregate (state and national) level data.

First, house price fluctuations are local outcomes and are specific to particular

economic areas, e.g. MSAs, see Baltagi and Li (2014). Second, MSAs in the sample are

subject to similar policy shocks (monetary policy, for example), taxes, and financial

market conditions. House prices at MSA level also exhibit much more fluctuations

both across space and over time than the smoother national or state level data can

provide, and this helps to exploit cross-sectional variation.

We begin with a standard dynamic panel analysis. The estimation results suggest

3See Anselin (1988) for details regarding spatial dependence and spatial heterogeneity.


that high fluctuations in house prices lower output growth. Our dynamic panel analysis

is related to that of Muñoz (2003) who examines the dynamics of US house prices using

state level data.

Next, a space-time model for house prices and output growth is specified. Using a

spatial connectivity weight matrix, the house price-output growth model is estimated.

Estimation results of the spatial model suggest that house price fluctuations have a

statistically significant negative effect on output growth. It is shown that the negative

effect of house price fluctuations on output growth are more pronounced during the

recent financial crisis.

As an alternative specification, we estimate the spatial model using a direct and

indirect effects approach. This is important because the recent literature in spatial

econometrics points out that standard estimation of spatial econometric models may

lead to misleading inference (LeSage and Pace 2009). Appropriate estimation involves

decomposition of spatial impacts into direct and indirect effects using a partial

derivatives impact approach. We decompose the impacts of house price fluctuations

on output growth into direct and indirect effects. It is shown that both the direct and

indirect impacts of house price fluctuations on real output are negative and

significant. House price fluctuation in a particular MSA, in addition to hampering its

own growth, transmits to neighboring MSAs.

Another major contribution of this paper is the application of a recursive estimation

of the house price spatial econometric model which provides an alternative measure

of house price synchronization. This technique enables investigation of the dynamics

of house price movements across space and over time where the spatial correlation

coefficient is allowed to vary over time and capture major changes in the economy. For

this purpose, we use a relatively longer time series of house price data. We consider

quarterly house price data for 373 MSAs during 1987:Q1 to 2014:Q3. The estimation

result shows that the spatial correlation coefficient across MSAs has been increasing

over time, indicating an increasing synchronization of house prices across MSAs during

the sample period.

The remainder of this paper is organized as follows. Section 2.2 presents a brief

summary of the literature review. Different existing theoretical and empirical studies

are discussed. Section 2.3 presents a space-time model for house prices and output

growth. Section 2.4 presents the data. Some stylized facts of the data are briefly


presented and discussed. Section 2.5 presents the empirical results, and the final

section provides the conclusion.

2.2 Brief literature review

Numerous studies on the interactions between house prices and the macroeconomy

have been conducted. Most studies focus on the temporal analysis of the interactions

between house prices and the real economy. Few studies have been conducted on the

relationship between house price fluctuations and real output in a space-time dynamic

framework. Studies that investigate the relationship between house price dynamics and

the real economy found that house prices play important role in the real economy and

show significant spatial patterns.

Cesa-Bianchi (2013) investigates the international spillovers of housing demand

shocks on the real economy. Using a global vector autoregressive model on 33

advanced and emerging economies over the period 1983 to 2009, finds that US house

demand shocks spill over to the real economy. Further, house demand shocks

originating from the US transmit to the other advanced economies. Using 379 US

metropolitan areas in a standard panel data model, Miller et al. (2011) investigate the

effect of house prices on output growth. They find that house price changes have

significant effect on output growth. Further, they show that the collateral effect

(change in actual consumption) of house prices has a stronger effect than the wealth

effect (change in desired consumption). Holly et al. (2010) employ an error correction

model with a cointegrating relationship between real house prices and real income

that explicitly considers heterogeneity and cross sectional dependence. Using 49 US

states during 1975-2003, they identify that real house prices rise in line with real

income and show significant spatial effects. Baltagi and Li (2014) replicate Holly et al.

(2010). First, they consider 381 MSAs instead of state level data. Second, they use

extended data during 1975-2011 instead of 1975-2003. They show that real house

prices and real income are co-integrated and the degree of spatial dependence is

stronger at MSA level than state level.

Iacoviello and Neri (2010), using a theoretical dynamic stochastic general

equilibrium (DSGE) model, study sources and consequences of fluctuations in the US

housing market. They find that slow technological progress in the housing sector

explains the upward trend in real housing prices. Over the business cycle, housing


demand and housing technology shocks explain one-quarter each of the volatility of

housing investment and housing prices.

Another direction of the literature in house prices has been the use of spatial

econometrics in standard hedonic price models. Anselin and Lozano-Gracia (2009)

briefly discuss the motivation and application of spatial econometric methods in

hedonic house price models. They state that there are two motivations for

incorporating spatial effects in standard hedonic models. The first is the need to

account for interaction effects and/or market heterogeneity. The second is to capture

spatial autocorrelation in omitted variables or unobserved externalities and

heterogeneities. Osland (2010) applies spatial econometrics on standard hedonic

house price models. Using municipality level data in the Southwestern part of Norway

during 1997 to 2002, the author shows that the spatial model alternatives have higher

explanatory power than the standard model. Clapp et al. (2002) use local polynomial

regression model to predict spatial pattern of house prices. They show that the local

polynomial regression model performs better in predicting the spatial pattern of

house prices across space.

In a more recent study, Dubé and Legros (2014) emphasize the importance of the

time dimension in spatial econometric estimation of hedonic house price models.

Using house price data in Paris between 1990 and 2001, they find that ignoring the

time dimension in spatial econometric estimation of hedonic house price models

could generate divergence in the estimated autoregressive coefficients. Can (1992)

formally considers spatial dependence and spatial heterogeneity in the standard

hedonic house price models. It is shown that models that include both spatial

dependence and spatial heterogeneity are superior to the standard hedonic house

price models. Using data for the year 1980 of 563 single-family houses sold in the

Franklin county of the Columbus metropolitan area, she finds significant spatial

effects in hedonic house price models.

2.3 Empirical methods

2.3.1 Model specification

Anselin (1988) states that spatial dependence in a regression framework reflects a

situation where the values of a variable at one location depend on the values of the


observation at other locations. A number of studies show that location is one of the

most important determinants of house prices, see, for example, Can (1992) and many

others.

Consider two neighboring MSAs i and j . Suppose the output growth process in a

particular MSA i at particular time period t is given by

gi t = f (g j t , vi t , v j t , gi t−1, xi t ), (2.1)

where gi t denotes the growth rate of per capita GDP for MSA i during time t , vi t

denotes the standard deviation of house prices as described in equation (2.6) below,

gi t−1 denotes the lagged output growth rate, xi t denotes a set of control variables,

unemployment, for example.

For a set of N MSAs i = 1, . . ., N , equation (2.1) can be written as

gi t = ρN∑

j 6=iWi j g j t +α1vi t +λ

N∑j 6=i

Wi j v j t +α2gi t−1 +α3xi t + c +εi t , (2.2)

or in matrix form

g t = ρW g t +α1vt +λW vt +α2g t−1 +α3xt + c +εt , (2.3)

where ρ is the spatial correlation coefficient, W is a spatial weight matrix connecting

MSAs i and j , α1, λ, α2, and α3 are unknown parameters, c is a constant, and εt is an

i .i .d white noise.

Equation (2.3) states that the growth regression relationship is between the N X 1

vector of time t growth rates (g t ), neighboring MSAs’ growth rate in the current time

period (W g t ), own volatility of house prices in the current time period (vt ), neighboring

MSAs’ volatility of house prices in the current time period (W vt ), growth rates in the

previous time period (g t−1), and set of controls, e.g. unemployment. The model in (2.3)

is known as the spatial Durbin (SDM) model.

The parameters of interest are ρ, α1, and λ. The parameter ρ measures the extent

of spatial dependence in the dependent variable. A positive value of ρ indicates that

output growth in neighboring MSAs affects a particular MSA’s growth rate positively. A

number of studies show that growth rates in neighboring units have positive effect on


the growth rate of a particular economic unit. Ertur and Koch (2007), for example, find

that the growth rates of neighboring countries play an important role in the growth rate

of a particular country through technological interdependence, see also Abate (2015).

The parameter α1 links the fluctuation of house prices in a particular MSA i to that

of the growth rate of output in that MSA itself. Different previous works show that an

increase in the fluctuations of house prices affects average growth rate negatively, see

Bordo and Jeanne (2002).

The effect of average house price movements from neighboring MSAs is measured

by the parameterλ. A high house price fluctuation observed in nearby MSAs might have

negative effect on the economic growth of a particular MSA while a relatively stable

house price changes in nearby MSAs may have positive effects on output growth rate of

a particular MSA. The temporally lagged growth rate is included in the model to account

for the fact that past growth may contain some information about the economy.

The other important model in spatial regression specifications is the spatial

autorgeressive (SAR) model of the form

g t = ρW g t +α1vt +α2g t−1 +α3xt + c +εt . (2.4)

This model is a special case of the model in (2.3) with λ = 0. This model states that

spatial dependence occurs through the dependent variable, see LeSage and Pace (2009)

for details as well as further discussion on the spatial error (SER) model where spatial

dependence occurs through the error terms. Setting the restrictions ρ = 0, and λ= 0 in

equation (2.3) produces the standard panel data specifications of the form

g t =α1vt +α2g t−1 +α3xt + c +εt . (2.5)

All the different spatial econometric models discussed above can be estimated

using maximum likelihood, instrumental variable estimation, and generalized method

of moments, see Elhorst (2010) for details.


2.3.2 Direct and indirect impacts

LeSage and Pace (2009) argue that appropriate estimation of spatial econometric

models such as in equations (2.3) and (2.4) involves decomposition of spatial impacts

into direct and indirect effects using the partial derivatives impact approach. Taking

the SDM in (2.3) as a point of departure, it can be rewritten as

g t = (I −ρW )−1(α1vt +λW vt +α2g t−1 +α3xt + c +εt ). (2.6)

The matrix of the partial derivatives of output growth, g t , with respect to an

explanatory variable, vt , for example, for all spatial units i = 1, ..., N is

[∂g t∂v1t

. . . ∂g t∂vN t

]= (I −ρW )−1

α1 w12λ . . . w1Nλ

w21λ α1 . . . w2Nλ

. . . . . .

. . . . . .

. . . . . .

wN 1λ wN 2λ . . . α1

.

The direct effect is the average of the diagonal elements, and the indirect effect is the

average of the off diagonal elements (LeSage and Pace 2009). The direct and indirect

effects approach here enables us to isolate the effects of house price fluctuations on

the real economy into direct and indirect effects. A surprise movement in house price

in a particular MSA may affect the growth rate in that MSA itself (direct effect) and

potentially affect the growth rate of other MSAs (indirect effect).

2.4 Data

This study covers 373 MSAs in the US during the period 2001 to 2013. The US Office

of Management and Budget (OMB) defines metropolitan areas based on a core area

containing a large population nucleus together with adjacent communities having a

high degree of economic and social integration with that core.

We draw data for house prices, per capita GDP, and unemployment from different

sources. All-transactions quarterly house price index for 373 MSAs from 2001 to 2013

is taken from the Federal Housing Finance Agency (FHFA). The all-transactions house

price index data of the FHFA is widely used in previous studies, see e.g. Bork and Møller


(2015), Baltagi and Li (2014) and Miller et al. (2011) among many others. The house

price indexes are constructed using repeated sales and refinancing on the same single-

family properties.4

The MSA level data are available on a quarterly level back to the mid-1980s. However,

per capita GDP data at the MSA-level is only available on a yearly basis. And, thus, to be

consistent with the per capita GDP data, we calculate house price volatility, vi t for MSA

i at particular year t as the standard deviation of log prices over four quarters for each

year as:

vi t = std .dev(log (pq1i , pq2i , pq3i , pq4i )), i = 1, ...,373; t = 2001, ...,2013 (2.7)

where pq1i , ..., pq4i is the house price index at each quarter for MSA i . Note that vi t is

normalized by the mean price of each MSA to control for size of each MSA.

The per capita gross domestic product (GDP) for each MSAs is drawn from the

Bureau of Economic Analysis (BEA) from 2001 to 2013. The metropolitan area GDP is

the sub-state counterpart of the Nation’s gross domestic product (GDP).

Unemployment data is collected from the Bureau of Labor Statistics (BLS). The

unemployment data for each MSA is available on a monthly frequency. The annual

unemployment growth rate is constructed using this monthly data for each MSA from

2001 to 2013.

Prior to the empirical analysis of house prices and output dynamics, it is of interest

to look at some features of the data. Table 2.1 presents the summary of the data for

GDP (in logs), GDP growth, growth rate in unemployment, and house prices (in logs).

Panel I of the table reports the descriptive statistics of the data across all MSAs from

2001 to 2013. The mean price volatility across all MSAs during 2001 to 2013 has been

above 2% per year. Growth rate of per capita GDP across all the MSAs has been above

0.5% per year during the sample period. Cross correlations of variables across all MSAs

during the sample period is reported in panel II. As shown, house price volatility has

an average negative correlation of -0.019 with output growth over the sample period.

Similarly, output level and house price fluctuations have an average negative cross

correlation of -0.090 during the sample period.

4See appendix A.2.1 for details.


Panel III of Table 2.1 reports metropolitan cities with highest and lowest house

price fluctuations and GDP growth rates. The highest mean volatility for the entire

period has been in Merced, California with average mean volatility of above 6%. The

lowest mean volatility has been observed in Cedar Rapids, Iowa where the mean

volatility has been around 0.65%. The highest mean real GDP per capita growth rate

have been observed in Corpus Christi, Texas with a mean real income growth rate

above 5.7%. The lowest mean income growth rate, on the other hand, have been

observed in the city of Canton-Massillon, Ohio where the mean growth rate for the

entire sample period has been around -2.2%.

Table 2.1: Data summary: 373 MSAs from 2001 to 2013

GDP GDP

growth

House

price

House price

volatility

Unemployment

growth

Panel I

Mean 10.564 0.0055 5.764 0.0209 1.803

Median 10.557 0.0045 5.729 0.0151 1.775

Std.dev 0.265 0.0357 0.179 0.0187 0.386

Panel II

GDP 1 0.1160 0.186 -0.0900 -0.1333

GDP growth 1 -0.064 -0.0192 -0.1466

House price 1 0.3250 -0.2166

House price volatility 1 0.0549

Unemployment 1

Panel III

Highest growth MSAs/value Corpus Christi, Texas/0.0577

Lowest growth MSAs/value Canton-Massillon, Ohio/-0.0216

Highest volatile region/value Merced, California/0.0628

Lowest variance region/value Cedar Rapids, Iowa/0.0065

Notes: Panel I reports the descriptive statistics of GDP, GDP growth, house price, house price volatility, and unemployment

growth. Panel II reports the cross correlations of GDP, GDP growth, house price, house price volatility, and unemployment

growth. Panel III reports metropolitan areas with highest and lowest GDP growth and house price volatility.


2.5 Results

2.5.1 Dynamic panel analysis

Prior to the empirical estimation, different panel unit root tests were performed. Levin

et al. (2002) and Im et al. (2003) are the popular panel data unit root tests in the

literature. However, both Levin et al. and Im et al. unit root tests are not valid unit root

tests in the presence of cross-sectional dependence. Baltagi et al. (2007), for example,

show that both the Levin et al. and Im et al. panel unit root tests can be biased in the

presence of cross sectional dependence.

Alternative panel unit root tests that allow for possible cross-sectional dependence

have been proposed in the literature. Pesaran (2007) suggests a panel unit root test that

allows for cross-sectional dependence where the standard augmented Dickey–Fuller

(ADF) regressions are augmented with the cross-section averages of lagged (CADF)

levels and first-differences of the individual series. Bai and Ng (2004) consider the

possibility of unit roots in the common factors where they apply the principal

component procedure to the first-difference version of the model, and estimate the

factor loadings and the first differences of the common factors.

In this paper, we consider Pesaran (2007) unit root test which is relevant to our

application. The null hypothesis of unit roots in output growth, house price volatility

and unemployment growth rate are rejected both with and without an intercept and a

trend, see Table 2.2. Note that the standard Levin et al. (2002) and Im et a. (2003) panel

unit root tests (not reported) also reject the null hypothesis of unit roots in output

growth, house price volatility and unemployment growth rate.


Figure 2.2: Simple correlation of output growth and house price volatility

Note. The figure shows plot of average output growth and average standard deviation of house prices across 373

MSAs during 2001-2013. The average growth and standard deviation of house prices for each MSA is computed over

the sample period 2001-2013. We report a sample of 40 MSAs for clarity. The pattern is more or less similar for all 373

MSAs.

Table 2.3 presents the maximum likelihood results of the dynamic panel

specification. The dependent variable is the annual growth rate of per capita GDP

computed as the log difference. The independent variables are volatility of house

prices measured as the standard deviation of log prices as defined in equation (2.6),

the unemployment growth rate, and previous growth rate as well as dummy for the

year 2007. Specifications in panel A are results for the whole sample period, whereas

specifications in panel B are results for the sub-sample period 2007-2013. Such

sub-period specification helps in understanding the interactions between changes in

house prices and output growth during the recent financial crisis.


Table 2.2: Pesaran’s panel unit root test results

With an intercept termCDAF(1) CDAF(2)

Volatility -2.470*** -2.553***GDP growth -3.211*** -3.267***Unemployment growth -1.807*** -1.851***

With an intercept and a linear trend termCDAF(1) CDAF(2)

Volatility -2.650** -2.711**GDP growth -3.460*** -3.549***Unemployment growth -2.569** -2.464**Notes: (***, **) denotes significance at (1%, 5%) level. The reported values

are the cross section averages of cross-sectionally augmented Dickey- Fuller

(CADF) test statistics, see Pesaran (2007).

Table 2.3: Dynamic panel results

Panel A: 2001-2013 Panel B: 2007-2013

(1) (2) (3) (4)

Constant 0.007 (0.001)*** 0.033 (0.003)*** 0.006 (0.001)*** 0.015 (0.005)***

Volatility -0.063 (0.027)** -0.069 (0.029)** -0.489 (0.039)*** -0.473 (0.042)***

Gr ow th−1 0.144 (0.015)*** 0.092 (0.016)*** 0.072 (0.019)*** 0.046 (0.020)***

Dummy for 2007 -0.012 (0.002)*** -0.016 (0.002)*** -0.003 (0.002) -0.005 (0.003)**

Unemployment growth -0.014 (0.002)*** -0.005 (0.002)**

Log likelihood 9347.64 8298.36 4920.06 4354.53

N 4849 4849 2611 2611

Notes: (***, **) denotes significance at (1%, 5%) level. Standard errors are in parenthesis. The dependent

variable is the change in (log) GDP per capita.

Column (1) of Table 2.3 reports the specification without unemployment growth rate.

The coefficient estimate of house price volatility shows a statistically significant

negative effect on the output growth rate. Figure 2.2 shows the scatter plot of average

volatility and average output growth rate over the entire period for a randomly selected

40 MSAs. The graph shows a clear negative relationship between volatility and output

growth.

A change in output might affect local industry structure and frictions in the labor

market and may cause migration of the labor force. To capture this effect, we include

unemployment growth rate as an additional control variable in column (2) of Table 2.3.


The effect of house price volatility on output increases (in absolute value) slightly. The

unemployment rate also takes a statistically significant negative coefficient estimate,

reflecting the standard relationship between unemployment and output growth.

Column (3) and (4) in panel B of Table 2.3 present the specification during the

sample period 2007-2013. Interestingly, the coefficient of volatility on output growth

shows an increase (in absolute) value during the period 2007-2013. This reflects that

the loss in output due to price fluctuations is more pronounced during crisis periods.

The remaining variables, past growth, and unemployment rate take predicted signs

across all specifications.5

Further, Figure 2.3 illustrates the dramatic changes in house prices and output

growth rate during the recent boom and bust of the housing markets. The figure

displays the median (blue line) of house price volatility with the first and third

quantiles (red lines) across the 373 MSAs during 2001 to 2013. The figure shows that

high price fluctuations are accompanied by lower output growth rates. This result

supports the inverse relationship between house price fluctuations and output growth

illustrated in Figure 2.1 in Section 2.1. Further, the figure shows that growth rate in per

capita GDP was at its lowest value in 2009.

Figure 2.3: Plots of price fluctuations and output growth across all MSAs during the period 2001-

2013

5Note that a complete analysis of house price volatility and output growth in a standard dynamic panel data framework isbeyond the scope of the present paper. The results presented in this section are for benchmark purpose for the empirical analysisin the next section. The commonly used method of estimation for dynamic panel data models is that of Arellano and Bond (1991)GMM approach. Estimation of house price fluctuations and output growth in our sample using the Arellano and Bond (1991) GMMapproach does not change our main qualitative conclusions.


2.5.2 Spatial modeling of house prices and the macroeconomic dynamics

In this section, we analyze the spatial dependence of house price fluctuations and

output growth rate across 373 US MSAs during the period 2001 to 2013. The section

starts with a discussion on the spatial weight matrix used in the empirical estimation.

The empirical analysis focuses on the two models given in equations (2.3) and (2.4).

The final section presents a time varying estimation of the spatial dependence.

A fundamental issue in the analysis of spatial econometric models in (2.3) and (2.4)

is the specification of the spatial weight matrix that defines a neighborhood structure.

More precisely, each MSA is connected to a set of neighboring MSAs by means of a

spatial pattern introduced exogenously in W . Elements wi j indicate the way MSA i is

spatially connected to MSA j . To avoid self neighborhood, the elements wi i on the main

diagonal are set to zero by convention.

There is little guiding theory in the selection of the appropriate weight matrix in

practice (Anselin 2002). Most commonly used weight matrices in spatial econometrics

are binary contiguity weight matrix, inverse distance weight matrix, and the k-nearest

neighbor weight matrices, see Anselin (1992). More complex spatial weight matrices

can be created based on additional theory and assumptions, such as those based on

economic distance (Holly et al., 2010). In this paper, we use a k-nearest neighbor row

normalized distance weight matrix. More specifically, the weight matrix in

standardized form is specified as

w(k)i j = w(k)∗i j /∑

w(k)∗i j wi th w(k)∗i j =

0 i f i = j

1 i f di j < di (k)

0 i f di j > di (k)

,

where di j is the great circle distance between metropolitan city centroids, and di (k) is

the k th order smallest distance between metropolitan city i and j so that each MSA

has k neighbors.6 In this paper, we consider k = 10.7 One advantage of choosing the

k-nearest weight matrix instead of the inverse distance weight matrix is that the latter

specification results in an unacceptably large number of neighbors for the smaller

units, see Anselin (2002).

6The great-circle distance, the shortest distance between any two points is determined as:di j = r adi ous x cos − 1[cos | l ong tui dei − long tude j | cosl ati tudei cosl ati tude j + si nl ati tudei si nl ati tude j ]. We

extrapolate the longitude and latitude coordinates for each MSA from the Census Bureau.7We also used different values of k (k = 6, 8, and 12) but the qualitative results are the same.


The estimation results of the spatial models are reported in Table 2.4. Panel A of the

table reports the full sample estimation results, and panel B reports the estimation

results of the period 2007-2013. In both samples, we estimate both the SAR and SDM.

The coefficient estimate of house price volatility shows significant negative effect on

output growth across all specifications. Particularly, house price fluctuations result in,

respectively, a 21.4% and 27.4% decline in output growth under the SDM SAR

specifications during the sample period 2007-2013. As discussed previously, changes

in house prices can have significant consequences on output through consumption

and investment spending. The spatial autoregressive coefficient (ρ) has a positively

significant coefficient estimate, suggesting growth spillover effects across MSAs in the

US during the sample period. Many empirical works, see e.g. Abate (2015), Ertur and

Koch (2007), and LeSage and Fischer (2008), document a positive significant growth

spillover effects across countries as well as regions.

Table 2.4: Spatial panel model results

Panel A: 2001-2013 Panel B: 2007-2013

SDM SAR SDM SAR

Constant 0.005 (0.002)** 0.007 (0.001)*** 0.001 (0.003) 0.006 (0.002)***

Volatility -(0.072) (0.036)** -0.045 (0.025)** -0.214 (0.051)*** -0.274 (0.038)***

Gr ow th−1 -0.085 (0.071) -0.731 (0.105)*** 0.0576 (0.019)** 0.051 (0.017)***

Unemployment growth 0.052 (0.016)*** -0.002 (0.001)*** -0.002 (0.001)** -0.002 (0.001)*

W ∗Volatility 0.029 (0.047) -0.133 (0.068)**

W ∗Gr ow th−1 0.109 (0.027)*** -0.009 (0.034)

W ∗Unemployment growth 0.0007 (0.001) 0.005 (0.002)**

ρ 0.564 (0.018)*** 0.579 (0.018)*** 0.537 (0.026)*** 0.549 (0.025)

Log likelihood 9754.22 9745.59 5113.65 5109.00

Wald test ρ = 0 944.51 (0.000) 1061.81 (0.000) 424.77 (0.000) 479.10 (0.000)

Log likelihood ratio 17.26 (0.001) 9.29 (0.000)

N 4849 4849 2611 2611

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in parenthesis. P-values are in parenthesis for

the log likelihood ratio tests. The dependent variable is the change in (log) GDP per capita.

2.5.3 Alternative regression frameworks

The empirical analysis so far suggests a negative relationship between output growth

and house price movements. The loss of output due to price fluctuations tends to be


higher during the recent financial crisis. In this section, we reexamine the overall

robustness of the main results. We first consider a direct and indirect impacts

approach following LeSage and Pace (2009). We then consider fixed effects alternative

regression frameworks to account for MSA specific characteristics that may not be

captured by the explanatory variables.

2.5.3.1 Direct and indirect impacts

In this section, we estimate the direct, indirect, and total impacts of house price

movements on output growth. Table 2.5 reports the direct, indirect, and total impacts

estimation results implied by equation (2.6).

Table 2.5: Spatial panel model results: Direct and indirect effects

Panel A: 2001-2013 Panel B: 2007-2013

SDM SAR SDM SAR

Direct effect volatility -0.073 (0.030)** -0.048 (0.022)** -0.232 (0.042)*** -0.286 (0.033)***

Indirect effect volatility -0.012 (0.074) -0.062 (0.028)** -0.499 (0.108)*** -0.331 (0.045)***

Total effect volatility -0.085 (0.071) -0.110 (0.050)** -0.731 (0.105)*** -0.617 (0.073)***

Direct effect growth 0.052 (0.016)*** 0.072 (0.015)*** 0.061 (0.019)*** 0.054 (0.019)***

Indirect effect growth 0.289 (0.057)*** 0.092 (0.019)*** 0.049 (0.069) 0.062 (0.022)***

Total effect growth 0.341 (0.057)*** 0.164 (0.034)*** 0.110 (0.067) 0.116 (0.041)***

Direct effect unemployment growth -0.002 (0.001)** -0.002 (0.001)*** -0.002 (0.001)* -0.002 (0.001)

Indirect effect unemployment growth -0.001 (0.003) -0.003 (0.001)*** 0.007 (0.004)* -0.002 (0.001)

Total effect unemployment growth -0.003 (0.003) -0.005 (0.002)*** 0.005 (0.004) -0.004 (0.002)

ρ 0.565 (0.018)*** 0.579 (0.018)*** 0.537 (0.026)*** 0.549 (0.025)***

Log likelihood 9754.22 9745.59 5113.65 5109.00

Log likelihood ratio 17.26 (0.000) 9.29(0.009)

N 4849 4849 2611 2611

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in parenthesis for estimation

results. P-values are in parenthesis for the log likelihood ratio tests. The dependent variable is the change in (log)

GDP per capita.

Panel B of Table 2.5 reports the estimation results of the spatial models for the period

2007 to 2013. Both the direct and indirect effects of house price volatility are negative

and significant. The magnitude, however, has increased (in absolute value) compared


to the full sample period results. The loss of output from house price fluctuations during

the crisis period is more pronounced.

2.5.3.2 MSA fixed effects specification

In order to account for MSA specific features that may not be captured by the

explanatory variables, we re-estimate a spatial fixed effects model.

Table 2.6 reports the MSA specific fixed effects spatial model results. The full

sample results are reported in panel A, the results for the sub-sample period 2007-2013

are reported in panel B. As shown, house price volatility has a statistically significant

negative effect on output growth across all specifications under both sample periods.

We also estimate MSA specific fixed effects spatial model under a direct and indirect

effects approach. Both the direct and indirect effects of house price movements have a

statistically negative effect on output growth after controlling for MSA specific fixed

effects during the sub-sample period 2007-2013, see Table A.1 in the appendix.

Table 2.6: Spatial panel model results: MSA fixed effects

Panel A: 2001-2013 Panel B: 2007-2013

SDM SAR SDM SAR

Volatility -0.099 (0.043)** -0.062 (0.029)** -0.306 (0.061)*** -0.372 (0.046)***

Gr ow th−1 -0.044 (0.015)** -0.015 (0.014) -0.113 (0.019)*** -0.088 (0.017)***

Unemployment growth -0.018 (0.003)*** -0.012 (0.001)*** -0.020 (0.005)*** -0.009 (0.002)***

W ∗Volatility 0.057 (0.056) -0.099 (0.083)

W ∗Gr ow th−1 0.156 (0.028)*** 0.126 (0.035)***

W ∗Unemployment growth 0.013 (0.004)*** 0.019 (0.005)***

ρ 0.577 (0.018)*** 0.586 (0.018)*** 0.562 (0.025)*** 0.571 (0.024)***

Log likelihood 9952.85 9934.27 5347.95 5336.18

Wald test ρ = 0 1036.68

(0.000)***

1112.13

(0.000)***

502.03 (0.000)*** 558.60 (0.000)***

Log likelihood ratio 37.17 (0.000) 23.55 (0.000)

N 4849 4849 2611 2611

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in parenthesis. P-values are

in parenthesis for the log likelihood ratio tests. The dependent variable is the change in (log) GDP per capita.


2.5.4 Time varying space-time model results

The results presented so far outline a substantial change over time of the role played

by the network interactions in house price and the macroeconomic dynamics. This

suggests the need to examine spatial dependence of house prices over time. For this

end, a rolling windows recursive estimation is employed to capture the structural

changes in house price dynamics over time. Specifically, we estimate a space-time

model of the form

log pt = ρW pt +αpt−1 +εt . (2.8)

Quarterly house price data for 373 MSAs during 1987:Q1 to 2014:Q3 is used in the

recursive sample estimation.8 We use rolling windows of 10 quarters and row

normalized 10 nearest weight matrix. The rolling estimation resulted in 108 coefficient

estimates. The rolling estimates of the spatial correlation coefficient is reported (blue

line), together with 95% confidence bands (red lines), in Figure 2.4.

Figure 2.4: Recursive estimation results of log house prices across 373 MSAs during 1987:Q1 to

2014:Q3

8Because longer time series observation for MSA level per capita GDP is not available, only the dynamics of house pricesimplied by equation (2.8) is investigated in the recursive analysis using a longer time series data of house prices. This longer timeseries of observations is particularly important to address the dynamics of spatial correlation of house prices over a relatively longtime period.


The figure shows that the network coefficient (ρ) has been increasing over time.

Particularly in the mid 1990s the spatial correlation coefficient shows a substantial

increase, implying an increasing integration of house prices across US MSAs. This

reflects the enormous increase in the correlation of house prices in the US across

different states after the deregulation of interstate banking in the US during 1995 to

1999, see Landier et al. (2015) for details. Cotter et al. (2011) have also documented an

increasing trend in house price correlation across US cities during the real estate

boom.

The time varying spatial correlation coefficient captures the dynamics of house

prices both across MSAs and over time. This paper is the first to document an

increasing house price integration across US MSAs and over time using time varying

space-time econometric model.

2.6 Conclusion

This paper examines the interactions between house price fluctuations and output

growth rate across 373 MSAs in US over the period 2001-2013. In order to examine the

dynamics of house price fluctuations and output growth in the recent crisis period, we

use a sub-sample period of 2007-2013. We examine the dynamics of house prices and

output growth in standard panel data models as well as spatial panel models. The

paper adds to the literature on housing markets and the real economy in three

important dimensions: (a) it explicitly allows spatial lag variables (b) uses direct and

indirect effects estimation and (c) uses time varying spatial econometric model.

The standard dynamic panel results suggest a significant negative association

between a movement in house prices and output growth. The negative impact of

house price fluctuations on output growth is larger during the recent financial crisis.

Next, using a spatial weight matrix, we analyze the dynamics of house prices and

output growth by allowing spatial interaction effects. We consider spatial autoregressive

and spatial Durbin models. Estimation results of the spatial autoregressive and spatial

Durbin models show that spatially lagged house price movements and output growth

rates are very important in examining the interactions between housing market and

the wider economy. The negative effects of house price volatility on output growth gets

larger during the recent crisis.


As an alternative specification, we follow LeSage and Pace (2009) and use the direct

and indirect effects approach. The partial derivative impacts approach shows that

house price fluctuations have both direct and indirect negative effect on output

growth rate. This result has two important implications for stabilization policies. First,

achieving stable house prices helps to stabilize the wider economy. Second, nearby

economic units have important roles in stabilizing/destabilizing a given economy.

Moreover, in order to account for MSA specific factors that may not be captured by the

explanatory variables, we re-estimate a fixed effects model. The main results remain

the same after controlling for MSA specific characteristics.

Another major contribution of this paper is the recursive estimation of the house

price spatial econometric model. This method provides an alternative measure of

house price co-movements across metropolitan areas over time. For this purpose, we

use relatively longer time series house price data. We consider quarterly house price

data for 373 MSAs during 1987:Q1 to 2014:Q3. The estimation result shows that the

spatial correlation coefficient across metropolitan areas has been increasing over time,

indicating an increasing synchronization of house prices across MSAs during the

sample period.

This paper opens up an important research path in understanding the interactions

of the housing market and the macreoconomy. One possible direction of future work

can be investigating the channels through which house price volatility affects output

growth in a space-time dynamic framework. Housing market bubbles can also be

examined in a joint space-time effects specification.

2.7 References

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Anselin, L. (1988) Spatial econometrics: Methods and models. Dordrecht, Kluwer.

Anselin, L. (1992) Space and applied econometrics. Special Issue Regional Science and


Urban Economics, 22.

Anselin, L. (2002) Under the hood: Issues in the specification and interpretation of

spatial regression models. Agricultural Economics, 27, 247-267.

Anselin, L. (2003) Spatial Externalities and Spatial Econometrics. International

Regional Science Review, 2, 153–166.

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2.8 Appendix

A.2.1 Metropolitan Statistical Areas; definition and criteria

Metropolitan Statistical Areas (MSAs) are defined by the Office of Management and

Budget (OMB). Each metropolitan statistical area must have at least 10,000 inhabitants

in the urban center and adjust areas that are connected to the urban centers by

commuting.

The Federal Housing Finance Agency (FHFA) requires that an MSA must have at least

1,000 total transactions before it may be published. Additionally, an MSA must have had

at least 10 transactions in any given quarter for that quarterly value to be published.

A.2. 2 Spatial panel model results: Direct and indirect effects with fixed effects

Table A.1: Spatial panel model results: Direct and indirect effects with fixed effects

Panel A: 2001-2013 Panel B: 2007-2013

SDM SAR SDM SAR

Direct effect volatility -0.099 (0.035)** -0.065 (0.0267)** -0.327 (0.050)*** -0.390 (0.041)***

Indirect effect volatility 0.001 (0.092) -0.0856 (0.036)** -0.596 (0.128)*** -0.481 (0.059)***

Total effect volatility -0.099 (0.089) -0.151 (0.062)** -0.924 (0.127)*** -0.871 (0.093)***

Direct effect growth -0.032 (0.016)* -0.014 (0.016) -0.107 (0.021)*** -0.090 (0.019)***

Indirect effect growth 0.313 (0.057)*** -0.019 (0.021) 0.155 (0.069)* -0.1112 (0.028)***

Total effect growth 0.281 (0.059)*** -0.033 (0.037) 0.0481 (0.071) -0.202 (0.047)***

Direct effect unemployment -0.0176 (0.003)*** -0.013 (0.002)*** -0.019 (0.005)*** -0.009 (0.003) **

Indirect effect unemployemnt 0.007 (0.005) -0.017 (0.003)*** 0.017 (0.007)* -0.011 (0.004)**

Total effect unemployment -0.011 (0.005)* -0.029 (0.005)*** -0.002 (0.006) -0.020 (0.006)**

ρ 0.577 (0.018)*** 0.586 (0.018)*** 0.562 (0.025)*** 0.571 (0.024)***

Log likelihood 9952.85 9934.27 5347.95 5336.18

Wald test ρ = 0 1036.68 (0.000)*** 1112.13 (0.000)*** 502.03 (0.000)*** 558.60 (0.000)***

Log likelihood ratio 37.17 (0.000)*** 23.55 (0.000)***

N 4849 4849 2611 2611

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in parenthesis for estimation results. p-values are

in parenthesis for the log likelihood ratio tests.


A.2.4 List of metropolitan statistical areas

Abilene, TX

Akron, OH

Albany, GA

Albany, OR

Albany-Schenectady-Troy, NY

Albuquerque, NM

Alexandria, LA

Allentown-Bethlehem-Easton, PA-NJ

Altoona, PA

Amarillo, TX

Ames, IA

Anchorage, AK

Ann Arbor, MI

Anniston-Oxford-Jacksonville, AL

Appleton, WI

Asheville, NC

Athens-Clarke County, GA

Atlanta-Sandy Springs-Roswell, GA

Atlantic City-Hammonton, NJ

Auburn-Opelika, AL

Augusta-Richmond County, GA-SC

Austin-Round Rock, TX

Bakersfield, CA

Baltimore-Columbia-Towson, MD

Bangor, ME

Barnstable Town, MA

Baton Rouge, LA

Battle Creek, MI

Bay City, MI

Beaumont-Port Arthur, TX

Beckley, WV

Bellingham, WA

Bend-Redmond, OR

Billings, MT

Binghamton, NY

Birmingham-Hoover, AL

Bismarck, ND

Blacksburg-Christiansburg-Radford, VA

Bloomington, IL

Bloomington, IN

Bloomsburg-Berwick, PA

Boise City, ID

Boulder, CO

Bowling Green, KY

Bremerton-Silverdale, WA

Bridgeport-Stamford-Norwalk, CT

Brownsville-Harlingen, TX

Brunswick, GA

Buffalo-Cheektowaga-Niagara Falls, NY

Burlington, NC

Burlington-South Burlington, VT

Casper, WY

California-Lexington Park, MD

Cambridge-Newton-Framingham, MA

Canton-Massillon, OH

Cape Coral-Fort Myers, FL

Cape Girardeau, MO-IL

Carbondale-Marion, IL

Carson City, NV

Fargo, ND-MN

Farmington, NM

Flagstaff, AZ

Cedar Rapids, IA

Chambersburg-Waynesboro, PA

Champaign-Urbana, IL

Charleston, WV

Charleston-North Charleston, SC

Charlotte-Concord-Gastonia, NC-SC

Charlottesville, VA

Chattanooga, TN-GA

Cheyenne, WY

Chico, CA

Cincinnati, OH-KY-IN

Clarksville, TN-KY

Cleveland, TN

Cleveland-Elyria, OH

Coeur d’Alene, ID

College Station-Bryan, TX

Colorado Springs, CO

Columbia, MO

Columbia, SC

Columbus, GA-AL

Columbus, IN

Columbus, OH

Corpus Christi, TX

Corvallis, OR

Crestview-Fort Walton Beach-Destin, FL

Cumberland, MD-WV

Dalton, GA

Danville, IL

Daphne-Fairhope-Foley, AL

Davenport-Moline-Rock Island, IA-IL

Dayton, OH

Decatur, AL

Decatur, IL

Deltona-Daytona Beach-Ormond Beach, FL

Denver-Aurora-Lakewood, CO

Des Moines-West Des Moines, IA

Dothan, AL

Dover, DE

Dubuque, IA

Duluth, MN-WI

Durham-Chapel Hill, NC

East Stroudsburg, PA

Eau Claire, WI

El Centro, CA

Elizabethtown-Fort Knox, KY

Elkhart-Goshen, IN

Elmira, NY

El Paso, TX

Erie, PA

Eugene, OR

Evansville, IN-KY

Fairbanks, AK

Fayetteville-Springdale-Rogers, AR-MO

Fayetteville, NC

Flint, MI

Florence, SC

Florence-Muscle Shoals, AL

Fond du Lac, WI

Fort Collins, CO

Fort Smith, AR-OK

Fort Wayne, IN

Fresno, CA

Gadsden, AL

Gainesville, FL

Gainesville, GA

Gettysburg, PA

Glens Falls, NY

Goldsboro, NC

Grand Forks, ND-MN

Grand Island, NE


Grand Junction, CO

Grand Rapids-Wyoming, MI

Grants Pass, OR

Great Falls, MT

Greeley, CO

Green Bay, WI

Greensboro-High Point, NC

Greenville, NC

Greenville-Anderson-Mauldin, SC

Hanford-Corcoran, CA

Harrisburg-Carlisle, PA

Harrisonburg, VA

Hartford-West Hartford-East Hartford, CT

Hattiesburg, MS

Hickory-Lenoir-Morganton, NC

Hilton Head Island-Bluffton-Beaufort, SC

Hinesville, GA

Homosassa Springs, FL

Hot Springs, AR

Houma-Thibodaux, LA

Houston-The Woodlands-Sugar Land, TX

Huntington-Ashland, WV-KY-OH

Huntsville, AL

Idaho Falls, ID

Indianapolis-Carmel-Anderson, IN

Iowa City, IA

Ithaca, NY

Jackson, MI

Jackson, MS

Jackson, TN

Jacksonville, FL

Jacksonville, NC

Janesville-Beloit, WI

Jefferson City, MO

Johnson City, TN

Johnstown, PA

Jonesboro, AR

Joplin, MO

Kahului-Wailuku-Lahaina, HI

Kalamazoo-Portage, MI

Kankakee, IL

Kansas City, MO-KS

Kennewick-Richland, WA

Killeen-Temple, TX

Kingsport-Bristol-Bristol, TN-VA

Kingston, NY

Knoxville, TN

Kokomo, IN

Lafayette, LA

La Crosse-Onalaska, WI-MN

Lafayette-West Lafayette, IN

Lake Charles, LA

Lake Havasu City-Kingman, AZ

Lakeland-Winter Haven, FL

Lancaster, PA

Lansing-East Lansing, MI

Laredo, TX

Las Cruces, NM

Las Vegas-Henderson-Paradise, NV

Lawrence, KS

Lawton, OK

Lebanon, PA

Lewiston, ID-WA

Lewiston-Auburn, ME

Lexington-Fayette, K

Lima, OH

Lincoln, NE

Little Rock-North Little Rock-Conway, AR

Logan, UT-ID

Longview, TX

Longview, WA

Louisville/Jefferson County, KY-IN

Lubbock, TX

Lynchburg, VA

Macon, GA

Madera, CA

Madison, WI

Manchester-Nashua, NH

Manhattan, KS

Mankato-North Mankato, MN

Mansfield, OH

McAllen-Edinburg-Mission, TX

Medford, OR

Memphis, TN-MS-AR

Merced, CA

Michigan City-La Porte, IN

Midland, MI

Midland, TX

Milwaukee-Waukesha-West Allis, WI

Minneapolis-St. Paul-Bloomington, MN-WI

Missoula, MT

Mobile, AL

Modesto, CA

Monroe, LA

Monroe, MI

Montgomery, AL

Morgantown, WV

Morristown, TN

Mount Vernon-Anacortes, WA

Muncie, IN

Muskegon, MI

Myrtle Beach-Conway-N. Myrtle Beach, SC-NC

Naples-Immokalee-Marco Island, FL

Napa, CA

Nashville-Davidson–Murfreesboro–Franklin, TN

New Bern, NC

New Haven-Milford, CT

New Orleans-Metairie, LA

Niles-Benton Harbor, MI

North Port-Sarasota-Bradenton, FL

Norwich-New London, CT

Ocala, FL

Ocean City, NJ

Odessa, TX

Ogden-Clearfield, UT

Oklahoma City, OK

Olympia-Tumwater, WA

Omaha-Council Bluffs, NE-IA

Orlando-Kissimmee-Sanford, FL

Oshkosh-Neenah, WI

Owensboro, KY

Oxnard-Thousand Oaks-Ventura, CA


Palm Bay-Melbourne-Titusville, FL

Panama City, FL

Parkersburg-Vienna, WV

Pensacola-Ferry Pass-Brent, FL

Peoria, IL

Phoenix-Mesa-Scottsdale, AZ

Pine Bluff, AR

Pittsburgh, PA

Pittsfield, MA

Pocatello, ID

Portland-South Portland, ME

Portland-Vancouver-Hillsboro, OR-WA

Port St. Lucie, FL

Prescott, AZ

Providence-Warwick, RI-MA

Provo-Orem, UT

Pueblo, CO

Punta Gorda, FL

Racine, WI

Raleigh, NC

Rapid City, SD

Reading, PA

Redding, CA

Reno, NV

Richmond, VA

Riverside-San Bernardino-Ontario, CA

Roanoke, VA

Rochester, MN

Rochester, NY

Rockford, IL

Rocky Mount, NC

Rome, GA

St. Louis, MO-IL

Sacramento–Roseville–Arden-Arcade, CA

Saginaw, MI

St. Cloud, MN

St. George, UT

St. Joseph, MO-KS

Salem, OR

Salinas, CA

Salisbury, MD-DE

Salt Lake City, UT

San Angelo, TX

San Antonio-New Braunfels, TX

San Diego-Carlsbad, CA

San Jose-Sunnyvale-Santa Clara, CA

San Luis Obispo-Paso Robles-Arroyo Grande, CA

Santa Cruz-Watsonville, CA

Santa Fe, NM

Santa Maria-Santa Barbara, CA

Santa Rosa, CA

Savannah, GA

Scranton–Wilkes-Barre–Hazleton, PA

Seattle-Bellevue-Everett, WA

Sebastian-Vero Beach, FL

Sebring, FL

Sheboygan, WI

Sherman-Denison, TX

Shreveport-Bossier City, LA

Sierra Vista-Douglas, AZ

Sioux City, IA-NE-SD

Sioux Falls, SD

South Bend-Mishawaka, IN-MI

Spartanburg, SC

Spokane-Spokane Valley, WA

Springfield, IL

Springfield, MA

Springfield, MO

Springfield, OH

State College, PA

Staunton-Waynesboro, VA

Stockton-Lodi, CA

Sumter, SC

Syracuse, NY

Tacoma-Lakewood, WA

Tallahassee, FL

Tampa-St. Petersburg-Clearwater, FL

Terre Haute, IN

Texarkana, TX-AR

The Villages, FL

Toledo, OH

Topeka, KS

Trenton, NJ

Tucson, AZ

Tulsa, OK

Tuscaloosa, AL

Tyler, TX

Honolulu (’Urban Honolulu’), HI

Utica-Rome, NY

Valdosta, GA

Vallejo-Fairfield, CA

Victoria, TX

Vineland-Bridgeton, NJ

Virginia Beach-Norfolk-Newport News, VA-NC

Visalia-Porterville, CA

Waco, TX"

"Walla Walla, WA

Warner Robins, GA

Waterloo-Cedar Falls, IA

Watertown-Fort Drum, NY

Wausau, WI

Weirton-Steubenville, WV-OH

Wenatchee, WA

Wheeling, WV-OH

Wichita, KS

Wichita Falls, TX

Williamsport, PA

Wilmington, NC

Winchester, VA-WV

Winston-Salem, NC

Worcester, MA-CT

Yakima, WA

York-Hanover, PA

Youngstown-Warren-Boardman, OH-PA

Yuba City, CA

Yuma, AZ

CH

AP

TE

R

3SPACE-TIME MODELING OF ELECTRICITY SPOT PRICES



Niels Haldrup


Abstract

Using data for the Nord Pool power grid, we derive a space-time Durbin model for

electricity spot prices with both temporal and spatial lags. Joint modeling of temporal

and spatial adjustment effects is necessarily important when prices and loads are

determined in a network grid. By using different spatial weight matrices, statistical

tests show significant spatial dependence in the spot price dynamics across areas and

estimation of the model shows that the spatial lag variable is as important as the

temporal lag variable in describing the spot price dynamics. We decompose the price

impacts into direct and indirect effects and demonstrate how price effects transmit to

neighboring markets and decline with distance. A forecasting comparison with a

non-spatial model shows that the space-time model has improved forecasting

64

CHAPTER 3. SPACE-TIME MODELING OF ELECTRICITY SPOT PRICES 65

performance for 7 and 30 days ahead forecasts. A model with time-varying parameters

is estimated for an expanded sample period and it is found that the spatial correlation

within the power grid has increased over time. We interpret this to indicate an

increasing degree of market integration within the sample period.

Keywords: Electricity spot prices; Nord Pool; recursive estimation; space-timedependence; forecast comparison

JEL classification: C32; C33

Acknowledgments

We wish to thank four anonymous referees and the editor for comments and

suggestions which greatly improved the paper. We would like to thank seminar

participants of CREATES, Aarhus University and GeoDa Center for Geospatial Analysis

and Computation, Arizona State University for their helpful comments and

suggestions. Financial support from Center for Research in Econometric Analysis of

Time Series - CREATES (DNRF78), funded by the Danish National Research

Foundation is gratefully acknowledged.

3.1 Introduction

Whilst there is much research on the temporal dynamics of electricity spot prices

(see e.g. Efimova and Serletis 2014; Haldrup et al. 2010; Haldrup and Nielsen 2006;

Higgs 2009; Huisman and Mahieu 2003; Maciejowska and Weron 2015; Park et al.

2006), less attention has been paid to the role of spatial dynamics of electricity spot

prices. However, such dynamics are necessarily important when prices and loads are

determined in a network grid of power exchange.

A number of previous studies recognize the importance of spot price

interdependencies in a grid of electricity areas. Park et al. (2006), for example, point

out how US regional spot market prices are characterized by spatial price

interdependence. In particular, for highly interconnected transmission systems,

temporal demand and supply imbalances and possible transmission congestion may

result in spatial price dependence across markets. Measurement problems in spot


prices may also result in spot price spatial dependence. In deregulated electricity

markets, price competition among the different markets will result in high spatial price

dependence. This implies that a spot price observed in a particular market is

determined (in part at least) by what happens elsewhere in the system. When

forecasting spot prices in a given market, it is thus helpful to know if past and current

spot prices in other markets can improve forecasts. Joint modeling of space-time

effects can help investigating the dynamics of spot prices in integrated physically

connected markets and accordingly, a simultaneous space-time model of electricity

prices is called for.

In time series models, temporally lagged values of the dependent variable are often

included to describe the price dynamics. A similar motivation can be used to account

for spatially lagged variables in electricity spot price dynamics. In deregulated

electricity price markets with simplified zonal pricing system as in Nord Pool that we

are going to examine empirically, transmission congestion problems imply that power

flows from the low price area towards the high price area.1 This indicates that the spot

price of a particular area depends on the nearby market bidding area prices as well

implying the need to account for spatial interaction effects.

Despite the key importance of the spatial element in electricity price dynamics,

spatial econometric modeling of electricity prices is rare in the literature. An exception

is Douglas and Popova (2011) who estimate a spatial error model for twelve US spot

market regions and show that spatial patterns play a significant role in electricity price

dynamics. Congestion problems in the transmission system together with grid

networks provide the framework for spatial patterns of price dynamics. A feature of the

analysis in Douglas and Popova (2011) is that they consider spatial interactions among

the error terms, but not spatial interaction effects among the dependent variable and

the independent variables in their model. The spatial econometrics literature stresses

that ignoring spatial dependence in the dependent variable and/or in the independent

variables may result in biased and inconsistent coefficient estimates for the remaining

variables (see e.g. Elhorst and Yesilyurt 2014 and LeSage and Pace 2009). This is a

standard result in econometrics namely that if one or more relevant explanatory

variables are omitted from a regression equation, then in general the estimator of the

1In a simplified zonal pricing mechanism like the Nord Pool, zonal prices are determined on the marginal bid in that zone. Inthe nodal price system like the Pennsylvania, Jersey, Maryland (PJM), a nodal system delivers prices and dispatch at the nodes, seeBjørndal et al. (2013) for details.


coefficients for the remaining variables is biased and inconsistent. In contrast,

ignoring spatial dependence in the error terms, if present, will only cause a loss of

efficiency. Anselin (1988) also notes that when the focus of interest is to examine the

existence and strength of spatial interactions, a model that includes the spatial lag of

the dependent variable is more appropriate than a spatial error model. Elhorst (2010)

and LeSage and Pace (2009) also recommend a spatial Durbin model (SDM) that

incorporates the spatial lags of both the dependent and independent variables.

In this paper we propose a space-time model of Nord Pool daily electricity spot

prices, but in contrast to Douglas and Popova (2011), we derive and estimate a more

flexible SDM that encompasses spatial dependence both in the dependent and

independent variables of spot prices. Because the SDM nests the spatial error model as

a special case, error dependence is also accounted for in the variance-covariance

matrix. One of the key features of the SDM is that it produces unbiased coefficient

estimates, even when the true data generating process is a spatial lag or spatial error

model (see e.g. Elhorst 2010 and LeSage and Fischer 2008). This is because the SDM

nests the spatial lag and spatial error models as special cases.

For a spatial regression model, a change in the explanatory variable of a particular

unit not only affects the dependent variable of that particular unit itself (the direct

effect) but also the dependent variables in other units (i.e., the indirect/spatial

spillover effects). As a result, LeSage and Pace (2009) suggest a partial derivatives

impact approach because the standard point estimates of the spatial regression model

specifications may lead to inconsistent coefficient estimates. We use the partial

derivative impacts approach to decompose the price impacts into direct and indirect

effects. Another feature of the spatial Durbin model is the model’s ability to capture

such direct and indirect effects. This model does not impose prior restrictions on the

magnitude of the spatial spillover effects which is usually the main focus in empirical

spatial econometrics. In contrast, in the spatial error model, these spatial spillover

effects are set to zero by construction which indicates that the model is less

appropriate in applications, see Elhorst (2012) for details.

Daily spot prices from 13 bidding areas in the Nord Pool power market during the

period January 1, 2012 to August 31, 2014 are used in the empirical study. The daily

average price plays an important role in the Nord Pool power market since it serves as

a reference price for forward and future contracts and other derivatives. First, we


estimate the non-spatial electricity spot price model using standard ordinary least

squares (OLS). In order to capture weather effects on spot price dynamics, we include

temperature variables as additional controls. Unlike Douglas and Popova (2011), we

apply classic Lagrange Multiplier (LM) tests designed by Anselin (1988) and robust LM

tests designed by Elhorst (2010) in order to test whether spatial interaction effects

need to be accounted for in electricity spot price dynamics.

We consider different spatial weight matrices in the construction of the LM tests

and discuss in detail the different properties of matrices. The spatial weight matrices

we use are: a) a spatial weight matrix constructed from the transmission capacity of 13

bidding areas, b) a geographical contiguity weight matrix, and c) a float weight matrix.

The latter weight matrix is constructed from the observation that when the power

connection capacity across exchange areas allows a free float of power for a given hour,

then prices are identical across neighbor areas. On the other hand, when the capacity

is insufficient, congestion will occur and prices will differ, see e.g. Haldrup and Nielsen

(2006) and Haldrup et al. (2010). The weight matrix is constructed by calculating the

fraction of hours over the entire sample period where prices are identical and hence

indicates the fraction of hours with non-congestion. When a fraction is relatively high,

it indicates a connection that is relatively well connected in terms of power capacity.

On the other hand, a small fraction indicates that the connection is relatively often

subject to congestion. The weight matrix is a different way of measuring the

transmission capacity across regions. Hourly prices are used to determine whether

there is congestion or not via an indicator variable, whereas the prices being modeled

in the study are daily prices for each region and hence are not directly related to the

construction of the weight matrix. The classic and robust LM tests indeed indicate a

highly significant spatial dependence in spot prices under all the spatial weight

matrices specifications that we consider.

A general spatial Durbin model that incorporates temporal as well as spatial lags of

spot prices and weather variables is estimated using quasi maximum likelihood

estimation. We quantify the role of spatially lagged dependent and independent

variables in spot price dynamics. The joint space-time modeling of electricity spot

prices is believed to be important for different reasons. From a spot price modeling

perspective, it indicates that current and past spot prices in other markets are

important variables in determining current spot prices of a particular bidding market.


Thus, joint modeling of space-time effects in spot prices can help improve forecasts.

Giacomini and Granger (2004), for example, show that ignoring spatial correlation,

even when it is weak, leads to highly inaccurate forecasts. We conduct a forecasting

exercise and find that the space-time model has improved forecasting performance for

7 and 30 days ahead forecasts compared to the non-spatial model.

Finally, we recursively estimate a time varying coefficients spot price SDM and

examine the evolution of spot prices over time and across bidding markets and hence

can provide a time varying measure of the degree of spatial correlation. To fully exploit

the advantage of longer time series observation in the recursive estimation, we use

expanded average daily spot price data from January 1, 2000 to October 18, 2014 for 9

bidding areas in the Nord Pool for which we have a complete sample of price data. We

find that the spatial price correlation within the Nord Pool grid has been steadily

increasing over time which we interpret as a measure of an increasing degree of

market integration.

The remainder of the paper is organized as follows. Section 3.2 provides a brief

overview of the Nord Pool power market. Section 3.3 presents a spatial Durbin model

for spot electricity prices. Section 3.4 presents the data used in the empirical study

along with the spatial weight matrices and the estimation and forecasting results are

presented in Section 3.5. The final section concludes.

3.2 The Nordic Power System

The Nordic countries Denmark, Finland, Norway, and Sweden have deregulated

their power markets in the early 1990s and have cooperated to provide an efficient

power supply, see e.g. Nord Pool (2004) and Haldrup et al. (2010) for brief details. Nord

Pool Spot was established as a company in 2002 as the world’s first market for trading

power. Today, it is also the world’s largest market of its kind and provides the leading

market for buying and selling power in the Nordic and Baltic regions.

The Nord Pool Spot exchange area is divided into a number of bidding areas. In

2011, for example, the Nord Pool Spot market had four bidding areas in Sweden (SE1,

SE2, SE3, SE4), two bidding areas in Denmark (DK1, DK2), five bidding areas in

Norway (NO1, NO2, NO3, NO4, NO5); Estonia (EE), Finland (FI), Lithuania (LT), and

Latvia (LV) constitute one bidding area each.

The different bidding areas help efficient distribution of power within the


transmission grid and ensure that area market conditions are optimally reflected in the

price. If grid bottlenecks exist, bidding area prices (called area prices) may be different

and if there are no grid bottlenecks across neighboring interconnectors, there will be a

single price across the bidding areas. When there are constraints in transmission

capacity between two bidding areas, the power will always go from the low price area

to the high price area. This principle is based on the law of one price: the power flow

will move towards the high price area with excessive demand. This system also secures

that no market members are assigned privileges on any bottleneck which is an

important feature of a deregulated liberalized market.

In terms of generating capacity, the Nord Pool power is generated from different

sources. In 2012, for example, over 70% of power supply in Denmark was generated

from thermal plants and approximately 29% of power supply was generated from wind

turbines (see Nord Pool 2013). Over 43% of power supply in Sweden was generated

from hydropower while over 65% of power supply in Finland was generated from

thermal power and 95% of power supply in Norway was generated from hydropower

plants.

The Nordic market participants trade power contracts for next-day physical delivery

at the Elspot market and trading is based on an auction trade system for each hour of

the following day. Day-ahead power prices, known as Elspot, are determined based on

supply and demand for every hour the following day. In the empirical study, we will

focus on the daily price because this price is relevant for forward and future contracts

on the financial power markets.

In the Nord Pool power market, the balance between consumption and generation

of power is regulated (in real time) through the regulating power market (Elbas) which

is managed by the Transmission System Operators (TSO). If consumption exceeds

generation, the regulating power market ensures that one or more producers deliver

more electricity to the grid. When this happens, the TSOs buy more power from

producers of excess capacity. If generation of power exceeds consumption, the

regulating power market ensures that one or more producers reduce the generation of

electricity. When this happens, the TSOs sell power to the producers. The Elbas market

is separate from the spot market which is at focus of the present paper. See Nord Pool

(2013) for further discussions and details.


3.3 Spatial modeling of spot prices

Highly interconnected transmission systems, temporal demand and supply

imbalances and transmission congestion in electricity spot prices may result in spatial

price dependence across markets. Park et al. (2006), for example, point out that

because of limited storability and cross-grid transmission, price interdependency

among neighboring markets are the typical features of electricity spot prices.

Unobserved features such as production capacity and maintenance problems are also

likely to result in spatial spot price dependence. This implies that a spot price observed

at a particular market is determined by what happens elsewhere in the system.

Consider a spot price, pt , observed in three neighboring bidding markets, i − 1, i ,

and i +1.2 Because of the spatial proximity/and or interconnected transmission in the

bidding markets, it can be assumed that the spot price at time t in market i depends on

the spot prices at all three markets at time t −1, and the spot prices at two markets at

time t .

Suppose this dependence is captured by

pi , t =βpi , t−1 +ρ1pi−1, t +ρ2pi+1, t +γ1pi−1, t−1 +γ2pi+1, t−1 + c +εi , t . (3.1)

The first term on the right hand side of equation (3.1) is the first temporal lag of the

spot price in market i , the second term is the current spot price in market i−1, the third

term is the current spot price in market i +1, the fourth term is the first temporal lag of

spot price in market i −1, the fifth term is the first temporal lag of spot price in market

i +1, c is a constant and the last term is a white noise error process.

Under the assumption of no spatial price dependence among bidding markets (ρ1 =ρ2 = γ1 = γ2 = 0), and equation (3.1) produces the conventional autoregressive AR(1)

spot price process. In a highly interconnected transmission system with deregulated

markets like the Nord Pool, nearby market prices still affect each other.

Using a spatial connectivity weight matrix wi j connecting bidding markets i and

j ( j = i −1, i +1), we can aggregate (see also Giacomini and Granger 2004) the process

given in (3.1) as

pi t =βpi t−1 +ρi+1∑

j=i−1wi j p j t +γ

i+1∑j=i−1

wi j p j t−1 + c +εi t , (3.2)

2Note that we will use bidding markets and bidding areas interchangeably.


where ρ is a parameter measuring the strength of spatial (contemporaneous)

dependence between bidding markets, wi j is a spatial weight coefficient, γ is a

coefficient measuring lagged spatial dependence and εi t is a white noise error process.

It is clear from (3.2) that a spatial lag is a distributed lag (lag in space), rather than a

shift in a given direction like in the time series case. Here each spatial weight wi j to be

discussed later reflects the spatial influence of bidding market j on bidding market i .

Note that we consider temporal as well as spatial lags to be of first order for simplicity.

Equation (3.2) can be generalized (in matrix form) as

pt = ρW pt +βpt−1 +γW pt−1 +Ztθ1 +W Ztθ2 + c +εt , (3.3)

where pt is an N x1 vector of spot prices during the sample period time t , W is an

N xN spatial weight matrix connecting bidding areas i and j , β, θ1 and θ2 are

associated parameter vectors, Zt is a set of controls (e.g. weather conditions, time

dummies etc.) variables, and εt is a white noise vector error process. The model given

in (3.3) is known as the dynamic spatial Durbin model (SDM) as it includes the spatial

lags of both the dependent and independent variables, see also Debarsy et al. (2011).

In section 3.4 we will discuss the design of the weight matrix W in more details.

The spot price pt is related to spot prices in neighboring bidding markets in the

current time period W pt , the previous periods spot prices pt−1, previous periods spot

prices from neighboring bidding markets W pt−1, a set of control variables in the

current period Zt as well as a set of control variables from neighboring markets W Zt

which are thought to exert influence on current spot prices.

LeSage and Pace (2009) explicitly discuss a number of theoretical econometric as

well as economic motivations for incorporating spatial lag variables in a regression

framework. In our particular case, the model in (3.3) captures the possible spatial

interaction effects that may arise in the system grid.

One of the distinctive features of the SDM in (3.3) is that it nests various models as

a special case. Under the assumption of no spatial interactions, ρ = 0, γ= 0 and θ2 = 0,

produces the conventional spot price regression model. Imposing the restriction that

γ= 0 and θ2 = 0 produces the dynamic spatial autoregressive (SAR) model of the form

pt = ρW pt +βpt−1 +Ztθ1 + c +εt .


The SAR model contains linear combinations of the dependent variable as

additional explanatory variables but excludes the spatial lags of the independent

variables. This model assumes that exogenous factors (e.g. weather conditions and

previous periods spot prices) observed in neighboring areas do not have direct effect

on spot prices of a particular bidding market. In the standard spatial econometrics

literature, the restriction γ = 0 and θ2 = 0 is used to test the hypothesis whether the

SDM can be reduced to the spatial lag model.

Similarly, imposing the restrictions γ+ ρβ = 0, and θ2 + ρθ1 = 0, equation (3.3)

produces the dynamic spatial error model (SER) of the form

pt =βpt−1 +θ1Zt + (I −ρW )−1(c +εt ). (3.4)

These restrictions also allow to test the hypothesis whether the SDM can be reduced

to the spatial error model. The SER specification implies that spatial interaction effects

occur through spatial propagation of unobserved disturbances.

Consider the SER model in (3.4) rewritten as

pt =βpt−1 +θ1Zt + c ′+µt ,

where c ′ = (I −ρW )−1c, µt = (I −ρW )−1εt or µt = ρWµt +εt . This specification shows

that the scalar error process µi t in a particular bidding market i at time t is a weighted

average of the errors in neighboring bidding markets and its own local disturbance εi t .

Using (I −ρW )−1 = I +∑∞k=1(ρW )k , we can write µt as

µt = (I +ρW +ρ2W 2 + ...)εt .

If the error vector process εt is i .i .d , the variance-covariance matrix of the local

disturbance (see e.g. Kapoor et al. 2004) is given as

E(µtµ′t ) = σ2

ε(I −ρW )−1(I −ρW )−1′

= σ2ε[I +ρ(W +W ′)+ρ2(W 2 +W W ′+W ′2)+ ...].

The variance-covariance matrix implies that if | ρ |< 1, the equilibrium disturbances

are correlated with each other but closer neighbors are more correlated than distant

neighbors.


Douglas and Popova (2011) state that the SER model is more appropriate to model

electricity prices because it is relatively convenient to estimate using panel data sets.

As stated earlier, when the interest is to examine spatial interactions, a full model

specification of the spatial interaction process is more appropriate than the SER

model. The SDM which is more flexible than the SER model produces unbiased

coefficient estimates even if the true DGP is SER. This is because the SER model is

nested within the SDM, and as a result error dependence is accounted for the

variance-covariance matrix. In our empirical sections, a test on parameter restrictions

shows that both the SAR and SER models are rejected in favor of the SDM.

3.4 Data description and spatial weight matrices

3.4.1 Data

Daily spot market electricity prices for 13 bidding areas during the period January 1,

2012 to August 31, 2014 (a total of 12,662 observations) from the Nord Pool power

market are used. The data is from the Nord Pool ftp server. The spot market bidding

areas include four regions from Sweden (SE1, SE2 SE3, SE4), one region from Finland

(FI), two regions from Denmark (DK1, DK2), five regions from Norway (NO1, NO2,

NO3, NO4, NO5), and one region from Estonia (EE). See Figure 3.1 for locations of the

bidding areas. Data for the bidding areas of the Estonian-Latvian border, and the

Latvia-Lithuanian border are not included because the spot price data is incomplete

for the years 2012 and 2013.


Figure 3.1: Map of the Nord Pool bidding areas

Source: Nord Pool 2014

The data series are plotted in Figure 3.2 for each of the 13 bidding markets. The

daily prices in our sample are the averages of the 24 hourly prices (log transformed)

measured in Danish kroner (DKK) per MWh.3 It can be seen that, in general, the spot

prices show huge fluctuations.

Whereas bidding areas from Sweden (SE1, SE2, SE3, SE4), Norway (NO1, NO2, NO3,

NO4, NO5), and Finland (FI) tend to show similar spot price patterns, the bidding areas

in Denmark (DK1 and DK2) also show a similar pattern while the spot price pattern in

Estonia (EE) is rather different.3Effectively, the DKK/euro exchange rate remains constant within the sampling period due to a pegged exchnage rate policy

of the Danish National Bnak.


Figure 3.2: Daily spot prices for 13 bidding markets in Nord Pool

The descriptive statistics of the daily spot prices for each of the 13 Nord Pool spot

markets are reported in Table 3.1. The mean spot price for the 13 bidding markets is


more or less the same across different markets. The mean spot price ranges from its

maximum 5.680 in EE to its minimum 5.397 in NO3. The daily spot prices also show

similar patterns in standard deviation across different bidding areas. The lowest

standard deviation is observed in EE while the highest standard deviation is observed

in DK1.4 A wide range of unit root tests were conducted and they all strongly reject the

unit root hypothesis.

Average cooling degree days (CDD) and average heating degree days (HDD) that

capture daily weather effects in electricity spot prices are calculated using

approximate weather locations for each of the 13 bidding areas.5

Table 3.1: Spot price descriptive statistics in 13 Nord Pool bidding areas

SE1 SE2 SE3 SE4 FI DK1 DK2 NO1 NO2 NO3 NO4 NO5 EE

Min 4.005 4.005 4.005 4.005 4.005 NN NN 3.949 4.067 3.556 4.005 4.005 5.116

Mean 5.502 5.503 5.512 5.539 5.608 5.524 5.569 5.409 5.407 5.397 5.497 5.489 5.680

Max 6.607 6.607 6.624 6.624 6.624 8.087 6.624 6.568 6.568 6.568 6.607 6.607 6.835

Std.dev 0.315 0.315 0.321 0.318 0.306 0.439 0.375 0.351 0.331 0.369 0.311 0.307 0.196

3.4.2 Spatial weight matrix for spot prices

The specification of the spatial weight matrix W is crucial in spatial econometrics.

However, typically there is little guidance in the choice of the correct spatial weight

matrix in empirical applications. The usual tradition in constructing the spatial weight

matrix has been geographical distance. However, it is not obvious that geography is the

most relevant factor in spatial interactions between the economic units under

consideration. The weight matrix represents the influence process assumed to be

present in the network and hence the choice of the weight matrix should represent the

theory a researcher has about the structure of the influence of the processes in the

network, see also Leenders (2002).

4We have 3 negative prices (before log transformation) in DK1 and 2 negative prices in DK2 and treat them as missingobservations when taken in log terms.

5Weather Underground (http://www.degreedays.net/) provides worldwide cooling and heating degree days for many weatherlocations in the world. We used approximate city weather locations in the calculation of the CDD and HDD for each of the 13bidding areas.


Figure 3.3: Four hypothetical neighboring electricity bidding markets

In the spatial econometric model (3.2), each spatial weight wi j reflects the spatial

influence of bidding market j on bidding market i . Consider, for example, four

hypothetical neighboring bidding markets M1, M2, M3 and M4 displayed as in Figure

3.3. Bidding market M1 is neighbor to M2, M3, and M4 (considering first and second

order neighborhood) whereas bidding market M2 is also first order neighbor to

bidding markets M3 and M4. Then, a first order binary contiguity weight matrix W (1

if two bidding markets are neighbors to each other and 0 otherwise) and its square W 2

can be specified as

W =

M1 M2 M3 M4

M1 0 1 0 0

M2 1 0 1 1

M3 0 1 0 0

M4 0 1 0 0

and W 2 =

M1 M2 M3 M4

M1 1 0 1 1

M2 0 3 0 0

M3 1 0 1 1

M4 1 0 1 1

The weights are assumed to be non-stochastic and exogenously given with the

properties; (i) wi j Ê 0, (ii) wi j = 0 if i = j , for any i = 1, ..., N . The second property

implies that no bidding markets are considered neighbors to themselves. Note that the

square matrix W 2 reflects second order contiguity neighbors (that are neighbors to the

first order neighbors). Because second order neighbor to a particular observation i

includes observation i itself, W 2 has non zero diagonal elements, see LeSage and Pace

(2009) for details. Sometimes the weight matrix is normalized such that∑N

j 6=i wi j = 1,

for i = 1, ..., N .

In order to capture the electrical transmission capacity of bidding areas, we follow


Douglas and Popova (2011) to construct the transmission weight matrix. We construct

the transmission weight matrix in Table 3.2 using the transmission capacity available

for each of the 13 Nord Pool bidding markets from the Nord Pool spot website. The

elements of the weight matrix are row normalized transmission capacities (in

megawatts) available between each bidding market. If there is no transmission

capacity between bidding areas, the element of the weight matrix is zero. We assume

the transmission capacity available is constant over the sample period.

The transmission capacity between any two bidding areas (how much power can be

transmitted in the grid) captures the possible spatial interactions between these areas.

If the spot prices differ between two areas, then the transmission capacity across these

areas is fully utilized towards the area with the higher price. If the capacity between two

areas is not fully utilized the prices in these two areas will be equal.

Table 3.2: Transmission weight matrix for the 13 Nord Pool bidding areas


SE1 0 0.133 0 0 0.669 0 0 0 0 0 0.199 0 0

SE2 0.325 0 0.572 0 0 0 0 0 0 0.079 0.025 0 0

SE3 0 0.504 0 0.297 0.083 0.047 0 0.069 0 0 0 0 0

SE4 0 0 0.606 0 0 0 0.394 0 0 0 0 0 0

FI 0.328 0 0.362 0 0 0 0 0 0 0 0 0 0.309

DK1 0 0 0.425 0 0 0 0 0 0.575 0 0 0 0

DK2 0 0 0 1 0 0 0 0 0 0 0 0 0

NO1 0 0 0.409 0 0 0 0 0 0.427 0.036 0 0.128 0

NO2 0.194 0 0 0 0 0.299 0 0.478 0 0 0 0.029 0

NO3 0 0.400 0 0 0 0 0 0 0 0 0.600 0 0

NO4 0.750 0.250 0 0 0 0 0 0 0 0 0 0 0

NO5 0 0 0 0 0 0 0 0.872 0.128 0 0 0 0

EE 0 0 0 0 1 0 0 0 0 0 0 0 0

Bidding markets corresponding to columns and rows are from Sweden (SE1, SE2, SE3, SE4), Finland (FI), Denmark

(DK1, DK2), Norway (NO1, NO2, NO3, NO4, NO5) and Estonia (EE). The sources of the transmission capacities

is the Nord Pool website (http://www.nordpoolspot.com/).

For an hourly frequency of observations, if there is insufficient transmission

capacity between the two areas, bottlenecks occur and price differences will naturally

arise. The surplus area will have a lower price than the deficit area as more power is


available compared to consumption. Consider, for example, two bidding areas with

SE1 as a lower price area and SE2 a high price area. If no transmission lines were

available between the two areas, they would have different prices. Assume there is a

capacity of K megawatt (MW) available between SE1 and SE2. The price in SE2 would

then move towards a lower price due to additional supply and the price in SE1 would

move towards a higher price due to higher demand. The available transmission

capacity is used to level out price differences as much as possible.

When the power connection capacity across exchange areas allows a free float of

electricity for a given hour, then prices are identical across neighbor areas. On the

other hand, when the capacity is insufficient, congestion will occur and prices will

differ, see e.g. Haldrup and Nielsen (2006) and Haldrup et al. (2010). An alternative

weight matrix we consider is based on this observation. It is constructed by calculating

the fraction of hours over the entire sample period where prices are identical and

hence indicates the fraction of hours with non-congestion. When a fraction is

relatively high, it indicates a connection that is relatively well connected in terms of

power capacity. On the other hand, a small fraction indicates that the connection is

relatively often subject to congestion. The fraction in each cell in Table 3.3 represents

the fraction of hours where prices are identical between bidding markets to the total

number of hours in the sample period. SE1 and SE2, for example, have a fraction of

0.992 and hence indicates an exchange point with only little congestion and hence a

high degree of spatial dependence. We will refer to the weight matrix defined in this

fashion as a “float weight matrix”. Note that congestion/non-congestion is determined

via an indicator variable for hourly data, whereas the econometric model is formulated

for daily price observations. Hence, the weights and the price data are not directly

connected. The float weight matrix is an alternative measure of the transmission

capacity across regions.


Table 3.3: Float weight matrix


SE1 0 0.992 0.962 0.897 0.655 0.552 0.688 0.523 0.462 0.868 0.833 0.488 0.511

SE2 0.992 0 0.971 0.904 0.656 0.530 0.557 0.530 0.469 0.859 0.825 0.495 0.442

SE3 0.962 0.971 0 0.929 0.676 0.577 0.714 0.535 0.473 0.836 0.805 0.498 0.458

SE4 0.897 0.904 0.929 0 0.629 0.595 0.759 0.507 0.448 0.779 0.750 0.477 0.435

FI 0.655 0.656 0.676 0.629 0 0.405 0.498 0.373 0.332 0.574 0.562 0.344 0.721

DK1 0.552 0.530 0.577 0.595 0.405 0 0.789 0.385 0.395 0.483 0.467 0.362 0.276

DK2 0.688 0.557 0.714 0.759 0.498 0.789 0 0.409 0.385 0.603 0.587 0.382 0.346

NO1 0.523 0.530 0.535 0.507 0.373 0.385 0.409 0 0.892 0.492 0.483 0.897 0.229

NO2 0.463 0.469 0.473 0.448 0.332 0.395 0.386 0.892 0 0.435 0.425 0.822 0.208

NO3 0.868 0.859 0.836 0.778 0.574 0.483 0.603 0.492 0.435 0 0.935 0.460 0.379

NO4 0.833 0.825 0.805 0.750 0.562 0.467 0.587 0.483 0.425 0.935 0 0.453 0.370

NO5 0.488 0.495 0.498 0.477 0.344 0.362 0.382 0.897 0.822 0.460 0.452 0 0.212

EE 0.511 0.442 0.458 0.435 0.721 0.276 0.346 0.229 0.208 0.379 0.370 0.215 0

Bidding markets corresponding to columns and rows are from Sweden (SE1, SE2, SE3, SE4), Finland (FI), Denmark

(DK1, DK2), Norway(NO1, NO2, NO3, NO4, NO5), and Estonia (EE).

We also consider a contiguity weight matrix as an alternative specification where the

elements of the contiguity weight matrix are 1 if two bidding markets are neighbors to

each other and zero otherwise.

3.5 Estimation results and forecasting

3.5.1 Quasi-maximum likelihood estimation of the SDM

Any of the spatial econometric models we discussed in Section (3.3) can be

estimated by maximum likelihood (ML) (Anselin 1988), quasi-maximum likelihood

(QML) (Lee 2010), instrumental variables (IV) (Anselin 1988), generalized method of

moments (GMM) (Kelejian and Prucha 1999), or by Bayesian Markov Chain Monte

Carlo methods (Bayesian MCMC) (LeSage 1997). One advantage of QML estimators is

that they do not rely on the assumption of normality of the disturbances. One

disadvantage of the IV/GMM estimators is the possibility of ending up with a


coefficient estimate for the spatial autoregressive coefficient outside its parameter

space. Also, finding an appropriate instruments is an issue.

We use the quasi-maximum likelihood to estimate our SDM. Consider the SDM

pt = ρW pt +βpt−1 +γW pt−1 +θ1Zt +θ2W Zt + c +εt . (3.5)

Denote ψ = (δ′, ρ, σ2)′ and ς = (δ′, ρ, c ′)′ where δ = (β, γ, θ′1, θ′2)′. At the true value,

ψ0 = (δ′0, ρ0, σ20)′ and ς0 = (δ′0, ρ0, c ′0)′ where δ0 = (β0, γ0, θ01, θ′02)′. The likelihood

function of (3.5) is (Lee 2004)

lnL(ψ, c) =−N T

2ln(2π)− N T

2ln(σ2)+T l n | I −ρW | − 1

2σ2

T∑t=1

[ε′t (ς)εt (ς)

],′

(3.6)

where εt (ς) = S(ς)pt −βpt−1 −γW pt−1 −Ztθ1 −Zt W θ2 − c, and S(ς) = I −ρW .

The QMLEs ψ and c are the extreme estimators derived from the maximization of

equation (3.6). When the disturbances εt are normally distributed, ψ and c are the

MLEs. But when the disturbances εt are not normally distributed, ψ and c are QMLEs.

Lee (2010) and Lee and Yu (2008) show that the QMLEs have the usual asymptotic

properties including consistency, normality and efficiency for dynamic spatial

econometric models.

3.5.2 Empirical results and test for spatial interaction effects

Before we estimate the SDM given in (3.3), we estimate the non-spatial version of

equation (3.3) assuming ρ = 0, γ = 0 and θ2 = 0. The Schwarz loss, Akaike loss and

Hannan and Quinn’s phi measures all suggest that the 4th lag is the optimal temporal

lag length. Day-of-week dummies were also included as additional co-variates in the

model which may explain why 4 lags rather than 7 lags were chosen by information

criteria. Table 3.4 contains the OLS estimation results of model (3.3) without spatial

interaction effects. The coefficient of the first, second, and fourth temporal lag price are

positive and significant. The heating degree variable enters with a significant coefficient

estimate reflecting that electricity is a significant energy source for heating in the Nordic

countries.

In order to test whether spatial interaction effects need to be accounted for in

electricity spot price dynamics, we apply classic Lagrange Multiplier (LM) tests

designed by Anselin (1988) and the robust LM tests designed by Elhorst (2010). The LM


test statistics for spatial interaction effects among the dependent variable is known as

the spatial lag model. The LM test among the error terms, on the other hand, is known

as the spatial error model. Both the LM lag and LM error tests which are based on the

residuals of the non-spatial model are asymptotically distributed as χ2(1). These test

the null hypothesis of no spatial interactions against the alternative hypothesis of

spatial interactions. Anselin (1988) points out that since both tests can have power

against the other alternative, it is important to take account of possible spatial lag

dependence when testing for spatial error dependence and vice versa. The robust LM

test takes into account such misspecification of the other forms, see Anselin et al.

(1996) for technical details.

Table 3.4: Estimation results: The non-spatial model with tests for dynamics using

transmission, contiguity and float weight matrices

Model

Constant 1.721 (0.031)***

pt−1 0.574 (0.009)***

pt−2 0.069 (0.010)***

pt−3 -0.012 (0.010)

pt−4 0.038 (0.008)***

C DD 0.0001 (0.002)

H HD 0.001 (0.001)***

Mon 0.031 (0.007)***

Tue 0.204 (0.008)***

Wed 0.116 (0.008)***

Thu 0.126 (0.008)***

Fri 0.102 (0.008)***

Sat 0.082 (0.007)***

Transmission W Contiguity W Float W

LM test: no spatial lag 803.85 (0.000)*** 882.77 (0.000)*** 1942.49 (0.000)***

Robust LM test: no spatial lag 2020.85 (0.000)*** 2856.35 (0.000)*** 2579.09 (0.000)***

LM test: no spatial error 7.61 (0.0009*** 54.66 (0.000)*** 1.27 (0.259)

Robust LM test: no spatial error 1224.61 (0.0009*** 2028.22 (0.000)*** 637.88 (0.000)***

No. Obs. 12662 12662 12662

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in and parenthesis for model results and

p-values are in parenthesis for LM test results.


The LM test results under the different weight matrices namely the transmission,

contiguity and float weight matrices are reported in the lower panel of Table 3.4. Note

that all the different weight matrices are used in row normalized form. When using the

classic LM test under the transmission weight matrix, both the hypothesis of no

spatially lagged dependent variable and no spatially lagged error term must be

rejected. The robust LM tests also reject both the hypothesis of no spatially lagged

dependent variable and no spatially lagged error term. This indicates that the

non-spatial model is rejected in favor of either the spatial lag or spatial error model

implying the need to account for spatial interaction effects.6

When using the contiguity weight matrix, both the classic and the robust LM tests

reject the hypothesis of both no spatially lagged dependent variable and no spatially

lagged error term. The LM test results under the float weight matrix also produce more

or less similar results. Elhorst and Yesilyurt (2014) and LeSage and Pace (2009)

recommend that when both the classic and robust LM tests reject the non-spatial

model in favor of either to the spatial lag or spatial error model, one better adopts the

SDM. We, thus proceed to the estimation of the SDM.

Prior to the SDM estimation, it is of interest to examine the simple cross-correlation

of spot prices in the 13 bidding areas. Table 3.5 contains cross-correlations of the

residuals of the 13 bidding areas of the Nord Pool power market. The bidding areas

show an average cross-correlation of 0.694 between each other. The residuals of

bidding areas from Sweden (SE1, SE2, SE3, SE4), Norway (NO1, NO2, NO3, NO4, NO5)

and Finland (FI) show strong correlations between each other. This correlation also

captures the pattern of spot prices displayed in Figure 3.2. Residuals in bidding

markets from Denmark (DK1, DK2) on the other hand, show relatively weak

correlation with the above bidding markets but exhibit strong correlation between

themselves. Whereas the strongest cross-correlation of residuals is observed between

SE1 and SE2, the weakest cross-correlation of residuals is observed between DK1 and

NO3.

6Anselin (1988) illustrates that when both the classic LM lag and LM error tests give similar results, one better considers therobust LM specification tests.


Table 3.5: Cross-bidding market correlation (mean= 0.694) of spot prices


SE1 1 0.999 0.984 0.921 0.799 0.358 0.434 0.862 0.839 0.823 0.981 0.971 0.639

SE2 1 0.985 0.922 0.799 0.359 0.435 0.861 0.839 0.822 0.979 0.969 0.638

SE3 1 0.938 0.812 0.369 0.446 0.848 0.823 0.809 0.963 0.952 0.643

SE4 1 0.769 0.391 0.484 0.788 0.765 0.749 0.899 0.891 0.601

FI 1 0.354 0.398 0.678 0.661 0.637 0.781 0.776 0.728

DK1 1 0.865 0.290 0.294 0.277 0.348 0.339 0.305

DK2 1 0.353 0.346 0.334 0.422 0.414 0.341

NO1 1 0.979 0.963 0.888 0.894 0.566

NO2 1 0.959 0.867 0.873 0.575

NO3 1 0.853 0.858 0.543

NO4 1 0.988 0.624

NO5 1 0.626

EE 1

Column (1) of Table 3.6 shows the estimation results of the SDM when using the

transmission weight matrix. The estimated coefficient on the spatially lagged

dependent variable W pt is significant and expresses strong spatial dependence. This

result indicates some important implications in spot price modeling. From a spot

price modeling perspective it shows that current spot prices in other markets are

important variables in determining current spot prices of a particular bidding market.

Thus, joint modeling of space-time effects in spot prices can help improve forecasts.

From an econometric point of view, appropriate consideration of the spatial lag

variables can help avoid omitted variable bias problem. The difference found in the

coefficient estimates of pt−1, for example, in Tables 3.4 and 3.6 might reflect the size of

omitted variable bias problem.

The estimation results of the SDM reported in Table 3.6 show that estimates are

rather similar regardless of the choice of weight matrix. The estimation results strongly

support the hypothesis that a spot price observed at a particular market is partly

determined by what happens elsewhere in the system. This is rather intuitive since

highly interconnected transmission systems, temporal demand and supply

imbalances, price competition and transmission congestion in electricity spot prices

may result in spatial price dependence between markets as we have argued in

motivating the analysis. Unobserved features such as generating production capacity


and maintenance problems are also likely to result in spot price spatial dependence.

Table 3.6: Estimation results of the SDM

Model Transmission W Contiguity W Float W

Constant 0.661 (0.218)*** 0.434 (0.091)*** 0.422 (0.129)***

pt−1 0.379 (0.053)*** 0.409 (0.065)*** 0.391 (0.059)***

pt−2 0.159 (0.000)*** 0.158 (0.0213)*** 0.169 (0.013)***

pt−3 0.057 (0.016)*** 0.121 (0.019)*** 0.069 (0.014)***

pt−4 0.124 (0.010)*** 0.125 (0.012)*** 0.138 (0.008)***

HDD 0.0001 (0.001) -0.0001 (0.000) -0.00008 (0.001)

C DD 0.0013 (0.002) -0.0004 (0.001) 0.002 (0.002)

Mon 0.006 (0.009) 0.008 (0.009) 0.007 (0.010)

Tue 0.069 (0.029)** 0.051 (0.016)** 0.046 (0.023)**

Wed 0.029 (0.016)** 0.028 (0.011)* 0.021 (0.013)

Thu 0.044 (0.025)** 0.031 (0.012)*** 0.028 (0.021)

Fri 0.036 (0.022)** 0.025 (0.009)** 0.022 (0.018)

Sat 0.029 (0.017)** 0.021 (0.010)** 0.019 (0.014)

W ∗pt (ρ) 0.643 (0.074)*** 0.765 (0.035)*** 0.773 (0.029)***

W ∗pt−1 -0.148 (0.087)* -0.269 (0.051)*** -0.242 (0.068)***

W ∗pt−2 -0.164 (0.011)*** -0.149 (0.027)*** -0.174 (0.019)***

W ∗pt−3 -0.064 (0.013)*** -0.128 (0.026)*** -0.073 (0.014)***

W ∗pt−4 -0.111 (0.017)*** -0.116 (0.013)*** -0.132 (0.015)***

W ∗HDD 0.0002 (0.001) 0.0003 (0.0001)** 0.0003 (0.001)

W ∗C DD -0.002 (0.001) 0.0003 (0.001) -0.002 (0.001)**

Wald test spatial lag 76.67 (0.000)*** 52.97 (0.000)*** 335.07 (0.000)***

Wald test spatial error 522.09 (0.000)*** 129.06 (0.000)*** 253.46 (0.000)***

R2 0.583 0.583 0.585

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in parenthesis for

estimation results and p-values for Wald tests.

One can perform a Wald test to examine if the SDM reduces to either the spatial

lag or spatial error model. The Wald test of restrictions on the SDM are reported in the

lower panel of Table 3.6. Accordingly, the hypothesis of the SDM can be simplified to

either the spatial lag or spatial error model is rejected by the Wald test, for all the weight

matrices considered.


3.5.3 Direct and indirect effects

The spatial spot price model in (3.3) provides very rich own and cross-partial

derivatives that quantify the magnitude of direct and indirect or spatial spillover

effects which arise from changes in bidding area i ’s characteristics such as weather

conditions and previous spot prices, for instance. A change in a single observation of

an explanatory variable will affect the bidding area spot price itself (the direct effect)

and potentially affect all other bidding areas indirectly (the indirect effect/spatial

spillover effects). The direct and indirect effects are the logical consequence of the

SDM, since the model takes into account other bidding markets dependent and

independent variables through the introduction of the spatially lagged dependent and

spatially lagged independent variables. In fact, LeSage and Pace (2009) note that the

ability of spatial regression models to capture these interactions represents an

important aspect of spatial econometric models.

Taking the SDM in (3.3) as a point of departure, it can be rewritten as

pt = (I −ρW )−1(βpt−1 +γW pt−1 +θ1Zt +θ2W Zt + c +εt ). (3.7)

The model formulation in (3.7) can be used to calculate the direct, indirect, and total

effects. The N xN matrix of partial derivatives of the spot price pt with respect to an

explanatory variable, pt−1, for example, for all spatial units i = 1, ..., N is

[∂pt

∂p1t−1. . . ∂pt

∂pN t−1

]= (I −ρW )−1

β w12γ . . . w1Nγ

w21γ β . . . w2Nγ

. . . . . .

. . . . . .

. . . . . .

wN 1γ wN 2γ . . . β

,

where wi j is the (i , j )th element of the weight matrix W . The direct effect is measured

by the average of the diagonal elements while the indirect (or spatial) spillover effect is

measured by the average of either the row or column sums of the non-diagonal

elements. However, the numerical magnitudes of the row and column sums of the

indirect effects are the same implying that it does not matter which one is used

(LeSage and Pace 2009 and Elhorst 2010). A general SDM model with k explanatory

variables leads to kxN 2 partial derivatives.


Table 3.7 reports the direct, indirect, and total effects estimation results of the

spatial Durbin spot price model. Because the direct and indirect effects are composed

of different coefficient estimates, LeSage and Pace (2009) suggest simulating the

distribution of the direct and indirect effects using the variance-covariance matrix

implied by the maximum likelihood estimates in order to draw inferences about the

statistical significance of the direct and indirect effects. We follow LeSage and Pace

(2009) and examine the aggregate direct and indirect effects to avoid interpretation

complications.

Since the direct and indirect effects results are similar when using each of the

different weight matrices, only the results for transmission weight matrix are reported.

To conserve space, we do not report the coefficient estimates of the dummy variables.

As shown in the table, both the direct and indirect effects of the first temporal lag

coefficient are significant. The significant negative indirect effect shows that nearby

prices spillover to closer bidding market regions.

Table 3.7: Effects decomposition of spot price dynamics

Model Direct effect Indirect effect Total effect

pt−1 0.408 (0.008)*** 0.235 (0.018)*** 0.643 (0.019)***

pt−2 0.142 (0.011)*** -0.149 (0.021)*** -0.008 (0.023)

pt−3 0.048 (0.010)*** -0.075 (0.025)*** -0.027 (0.029)

pt−4 0.115 (0.009)*** -0.075 (0.019)*** 0.039 (0.021)*

HDD 0.0005 (0.001)** 0.0007 (0.001)** 0.0012 (0.001)**

C DD 0.0008 (0.001) 0.001 (0.002]) 0.002 (0.003)

W ∗pt (ρ) 0.643 (0.006)***

No. Obs. 12662

Wald test spatial lag 369.29 (0.000)***

Wald test spatial error 1183.89 (0.000)***

Notes: *** (**, *) denotes significance at 1% (5%, 10%) level. Standard errors are in

parenthesis for model results and p-values are in parenthesis for Wald test results.

Transmission weight matrix is used in the direct and indirect effects estimation.

3.5.4 Forecasting performance

Accounting for spatial interaction effects in spot prices is important not only for

estimation accuracy and efficiency but also may provide better forecasting

performance. Incorporating past and current spot prices in nearby bidding markets


could improve the forecast accuracy of the joint space-time model compared to the

non-spatial model. In this section, we evaluate and compare the forecasting

performance of the joint space-time model with a model without spatial dependence

using a root mean squared forecast error (MSFE) criterion. The benchmark model is

essentially the model (3.3) without spatial interaction effects, that is, the model

estimated by OLS with ρ = 0, γ= 0, and θ2 = 0.

Fingleton (2014) and Baltagi et al. (2013) suggest a spatial approach for dynamic

forecasting. For both the non-spatial and the spatial models, we forecast h-days ahead

spot price for all the 13 bidding markets. In the first step, the model in equation (3.3) is

estimated. In the next step, the reduced form in equation (3.7) is used to generate the

forecasts.

In order to evaluate the forecasting performance of both the spatial and non-spatial

models, we divide the full sample into an initial estimation period from July 31, 2012

to August 31, 2013 and a forecasting period covering 1 September 1, 2013 to August 31,

2014. For all the 13 bidding markets, we perform a 1 day, 7 days (one week) and 30 days

(one month) ahead recursive forecasts. The parameter estimates are obtained using an

expanding window where the estimation period increases by one observation when we

move one step ahead in time. For the spatial model, the estimation results using the

transmission weight matrix are reported.

Table 3.8: Forecasting evaluation

Forecast horizon

1 7 30

RMSFE

Non-spatial model 1.689 8.512 12.501

Spatial Durbin model 1.705 8.308 12.189

The results of the forecasting exercise are reported in Table 3.8. The prediction

performance is measured by sum of the root mean square forecast error (RMSFE) for

the single prices. The RMSFE for each model is computed for all the 13 bidding

markets over the forecasting period. As seen, the spatial model produces better

forecast accuracy particularly for one week ahead and one month ahead prediction

horizons. For one day ahead predictions, the forecast performance of the spatial and


non-spatial models is rather similar.

3.5.5 A time-varying coefficients SDM

The Nordic power grid and the associated power market has experienced significant

deregulation over the past 15 years. This concerns both the design of the auction market

conditions and improvements in the physical power transmission system. The purpose

of such deregulation and liberalization has been to improve the general competitive

market environment for power. Intuitively, such deregulation should increase spatial

price correlation across power grid points and hence considering the spatial correlation

fixed for a long sample period is questionable.

In this section, we estimate our SDM using recursive estimation to examine the

evolution of the coefficient estimates of the spot price SDM over time and with

particular focus on the spatial correlation. To this end, we use a somewhat longer time

series for spot prices data that covers the period January 1, 2000 to October 18, 2014.

Because longer time series observations of data are not available for all 13 bidding

markets, we consider only 9 bidding markets (a total of 48,645 observations) for which

we have daily spot price data covering the entire sample period. We use one bidding

market in Sweden, five bidding markets in Norway, the two bidding markets in

Denmark and one bidding market in Finland. We employ 2 months rolling window

recursive estimation of the SDM. In the first sample period considered for analysis in

this paper, we intend to cover as many bidding market regions as possible. For

recursive estimation, we intend to cover a longer span of time series observations in

order to address the development of the spatial correlation over a relatively longe time

period.

The evolution of time-varying coefficient of the spatially lagged dependent variable

implied by the spatial Durbin model is displayed in Figure 3.4. A transmission weight

matrix is used in the recursive estimation. The recursive estimates of the spatial

correlation coefficient are shown (red line), together with 95% confidence bands (blue

lines).

At the beginning of the sample, the graph shows an increasing spatial correlation.

The estimated coefficient exhibits a sharp fall around January 2010 when the Nordic

power market experienced extreme spikes in power prices. To better understand the


Figure 3.4: Time varying spatial correlation coefficient

dynamics of the time-varying spatial correlation coefficient, we plot the average spot

price dynamics across 9 bidding markets in Figure 3.5 and the spot price dynamics for

each of the 9 bidding markets in Figure 3.6.

Figure 3.5: Plot of average price across 9 bidding markets in the Nord Pool

As shown in Figure 3.5, the highest price peaks occurred around January 2010 over


Figure 3.6: Plots of spot prices across 9 bidding markets in the Nord Pool

the sample period. Figure 3.6 also shows that spot prices in many of the bidding

markets exhibit extreme spikes during January 2010. Specifically, with the exception of

the bidding markets DK2, NO1, and NO2, all other bidding markets experienced

extreme spot price spikes around January 2010. In fact, these price spikes resulted in a

general debate on the functioning of the power market that consequently were

scrutinized by the regulatory authorities, see e.g. NordReg (2011).

The sharp fall in the estimated spatial correlation coefficient in Figure 3.4 can be

caused by these unusual extreme spot price spikes. Notwithstanding, it is obvious that

over the sample period considered, there has been an increasing trend in the spatial

correlation within the Nord Pool grid. Without referring to an exact structural model,

we interpret this empirical finding as evidence of increased market integration and

competition across the bidding areas within the sample period. This is in line with the

NordReg (2013) findings where, using a number of different market indicators such as


the number of suppliers, the supplier switching rate, the price differences in the retail

markets, and the concentration in the wholesale market, the Nordic electricity markets

appear to have become increasingly competitive.

3.6 Conclusion

Spatial panel econometric models are becoming increasingly important to describe

many observed economic phenomena. We use tools from spatial econometrics to

examine the spatio temporal patterns of electricity spot prices within the Nord Pool

power grid. A dynamic spatial Durbin model that incorporates space-time effects in

the dynamics of electricity spot prices is developed.

The analysis shows that spatial dependence is significant and improves forecasting

performance over longer horizons. Also, we have descriptive evidence that the degree

of spatial correlation has increased in the sample period which may be interpreted as

evidence of increased market integartion and competion.

This paper opens up for some future research directions in electricity price

modeling and forecasting. It is obvious from the empirical findings of the current

paper that spatial effects are extremely important in describing the electricity price

dynamics. However, when moving on to analyze high frequency hourly electricity

price data (rather than daily average prices), the possibility of congestion and

non-congestion episodes across regions becomes important. The building of

empirical models that can capture such (spatial) regime switching price behavior is a

challenging modeling task that can contribute further to better understand the

complex spatio-temporal dynamics of power prices.

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Essays in Spatial Econometrics - AU Pure · econometrics have provided a powerful tool for examining spatial dynamics across different economic units. In contrast to standard econometrics,